Download EQUATIONS and INEQUALITIES

RADICALS AND SOLVING QUADRATIC EQUATIONS
Evaluate Roots
Overview of Objectives, students should be able to:
1. Evaluate roots
a. Simplify expressions of the form
2
a2
Main Overarching Questions:
1.
How do you simplify radical expressions with variables and constants?
2. What is the difference between the rules for even roots and odd roots?
b. Simplify expressions of the form 3 a 3
2. Evaluate higher order even and odd roots
a. Simplify expressions of the form
n
Objectives:
•
Simplify expressions of the form
an
Activities and Questions to ask students:
2
a2
•
If necessary review the basics of simplifying square root expressions found on pgs. 76-77 of
the MAT 0002/0024 guidebook.
•
In short: what is
•
draw to the conclusion that a 2 = a . Ask students to give a verbal description of this rule
(i.e. “if we take the square root of a squared quantity, the result is the original quantity not
squared.”)
To then extend the rule to negatives, go back to the original example, and ask students what
9 ? What is
3 2 ? After a few more examples of this kind, have students
(−3) 2 is. If they assume there is NO SOLUTION, ask them to follow order of operations to
simplify. What is the result? Was the result the same as the original quantity being squared
in the radical? What happened? Is the result similar to the original quantity being squared?
•
•
•
Does our original rule a 2 = a hold? Why or why not?
Repeat this type of example a couple of times. What pattern do you notice? What happens
to the quantity being squared inside the radical each time? What is the sign of the result?
Once students notice the result is always the positive of the number being squared inside the
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radical, ask them to think about what operator we have studied that “always makes things
positive.” They may consider using a negative sign, but remind them we want to use the
same operator for all numbers.
Have students draw to the new conclusion that
•
If students need or would like to practice at this point give a few problems from the following
Simplify expressions of the form
3
a3
( x − 1) 2 , − 49x 6 ,
(−10) 2 ,
(easiest to hardest):
•
a2 = a .
•
x 2 + 12 x + 36
•
If necessary review the basics of simplifying cube root expressions found on pgs. 76-77 of the
MAT 0002/0024 guidebook.
•
In short: what is
•
draw to the conclusion that 3 a 3 = a . Ask students to give a verbal description of this rule
(i.e. “if we take the cube root of a cubed quantity, the result is the original quantity not
cubed.”)
Hopefully, the students will ask about the negative cases. Go back to the original example,
3
8 ? What is
3
2 3 ? After a few more examples of this kind, have students
and ask students what 3 (−2) 3 is. If they assume there is NO SOLUTION, ask them to follow
order of operations to simplify. What is the result? Was the result the same as the original
quantity being cubed in the radical? What happened? Is the result similar to the original
quantity being squared?
•
•
•
•
Does our original rule 3 a 3 = a hold? Why or why not?
Repeat this type of example a couple of times. What pattern do you notice? What happens
to the quantity being squared inside the radical each time? What does the sign of the answer
depend on?
How is this rule different from the square root case?
If students need or would like to practice at this point give a few problems from the following
(easiest to hardest):
•
Simplify expressions of the form
n
an
•
•
•
3
(−100) 3 ,
3
( x + 1) 3 , − − 125x 3
We want the students to establish a rule for general nth degree roots.
Students should have noticed that with square roots we needed an absolute value bar to
ensure variable expressions were always positive. For cube roots we should not include the
bars because negatives are okay.
What about 4th roots? Students have not been introduced to roots above the cube root, so
ask them what they think a
4
radical does. If they are unsure have them consider how
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square roots and cube roots operate and draw a conclusion about 4th roots and higher. If
•
•
necessary give them the example that 2 4 = 16 and then 4 16 = 2 . Have them state the
process in words.
Repeat this process for 5th roots and higher.
Ask students to use the process from the previous two discussions to develop the rules for
what 4 a 4 = and 5 a 5 = . If necessary, give numeric examples to begin the discussion but
the analysis should be similar.
a 4 = a and
a5 = a
•
Have students draw to the conclusion that
•
Ask the students if they see a pattern. Can they generalize these rules to all roots? What
seems to be happening for even roots? Odd roots?
•
Have students draw to the conclusion that if n is even:
•
•
Have students draw to the conclusion that if n is odd: n a n = a
For practice use problems that are of a similar type given above.
4
n
5
an = a
Multiply Radical Expressions
Overview of Objectives, students should be able to:
1. Use the product rule to multiply radicals
2. Use factoring and the product rule to simplify
radicals
3. Multiply radical expressions and then simplify
4. Multiply radical expressions with more than one
term.
5. Use polynomial special products to multiply radicals
Objectives:
•
Use the product rule to multiply radicals
Main Overarching Questions:
1.
How do you know when radical expressions can be combined through multiplication?
2. How can you decide which rules to use when simplifying radical expressions?
Activities and Questions to ask students:
•
The product rule has already been established for square roots on pg 77 of the MAT
0002/0024 discussion. However, given the importance of this rule in both multiplication AND
simplification, it is probably worth a second discussion.
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Ask students to evaluate:
•
•
Ask students to evaluate:
Have students work several pairs of examples of this type. Does a pattern exist?
•
•
What is a 2 ⋅ b 2 ? What about
square root process is.
Do you notice a pattern?
•
•
Have students draw the conclusion that: a ⋅ b = a ⋅ b
Have students work several examples with higher order roots to establish:
n
•
•
•
Use factoring and the product rule to simplify
radicals
4⋅ 9.
4 ⋅ 9 . Do you see a relationship?
•
•
a 2 ⋅ b 2 ? Remind students to think about what the
a ⋅ n b = n a ⋅b .
Ask students if they see any limitations to the rule. Can we multiply any radicals? What
requirements does the rule establish? If students have trouble, ask them if we can apply the
rule to this problem: 2 x ⋅ 3 4 x . Why or why not?
Give students a few problem to multiply (they may not know how to simplify; this will be
introduced in the next concept)
This concept was also introduced in MAT 0024 but in simpler types of problems. As a
review:
•
How would you simplify 18 = 9 ⋅ 2 using the product rule? Describe the process. When
breaking up 18 to 9 ⋅ 2 , what process are you using? Why would you choose to factor 18
into 9 ⋅ 2 instead of another choice like 3 ⋅ 6 ? How would you know how to factor?
Summarize the simplifying process. Students should see the goal is to factor the radicand
such that one of the factors is the greatest possible perfect square.
•
•
How about 2 12 ? How does the “2” on the outside change the process?
Have students work several numeric examples of this type as a review.
6
•
Start with an example like x . Ask students to simplify the radical. Give several examples with
even exponents for students to work.
• How is this like what we have done before? If necessary remind students of what rules have been
established so far for radicals.
• What about x = x ⋅ x ? Do you see a pattern or process to if the exponent is odd? Give
students several more examples to work.
7
•
6
1
If necessary, give a summary of problems of the following types (easiest to hardest):
50 ,
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72x 2 , 100 x 3 y 9 , 36( x + 2) 2 ,
•
Multiply radical expressions and then simplify
3x 2 − 6 x + 3
•
•
Now, the factoring and simplifying process will be extended to higher order roots.
When simplifying square roots we wanted to factor the radicand into the greatest possible
perfect square. Ask students how they would handle cube roots. What should we factor our
radicand into?
•
Let students discuss some numeric higher order examples:
•
Start with an example like
•
What about
•
Next, x . What is the result? Do you see a pattern with the cube root? How does the power of
radicand change when taking the cube root? Do you see a pattern?
•
Next, give students a simple example to simplify:
x = x ⋅ x . Ask students to explain a way to
simplify cube roots where the power of the radicand is not a multiple of three.
•
Give a few examples to work:
•
Give the process for simplifying square roots and cube roots, ask students to conjecture how they
would handle higher order roots. If they have trouble, say: “For square roots we divided even powers
in the radicand by 2 to take the root, for cube roots we divided powers that are multiples of 3 by 3,
th
th
what do you think we would do with 4 , 5 , etc roots?” What must the powers in the radicand be
multiples of in case? This is usually a difficult concept for students to see since there are so many
different types of problems, so at this point, ask them to summarize the process for simplifying roots
in general and add specifics for each type of root.
•
Have students use their process to work the following types of problems:
•
This concept is a combination of the multiplication and simplifying concepts just introduced.
3
3
54 ,
3
− 16
x 3 . What is the result?
x 6 . Students may have some trouble with this. Ask them what the cube root does.
6
So, what quantity cubed gives us x ?
3
3
9
3
3
y 10 ,
3
4
3
3
1
( x + y ) 5 , 3 32 x 9 y 17
5
y 17 , 4 96x11
To facilitate the discussion give students a problem to look at: 50 xy ⋅ 4 xy 2 . Ask them
to multiply and simplifying the expression. Then ask them to summarize the process they
used in words, highlighting the major steps (i.e. combine the two radicals through
multiplication, factor each quantity in the new radicand, take the square root, and then
recombine any remaining radicals into one radical).
•
50 xy ⋅ 4 xy 2 = 10 x ⋅ y 2 y
Next, give them the same problem, with the added condition that all variables are considered
to be positive quantities. Many of them will think a major change is necessary, but ask them
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to look at their final answer. If all variables are positive, what operator could be removed?
•
Why could the absolute value bars we removed? 50 xy ⋅ 4 xy 2 = 10 xy 2 y . From now
on, all variables will be assumed to be positive removing the need to include absolute value
bars.
Give students several problems to practice with of varying difficulty level (square roots, cube
roots, and higher order roots). Include at least one problem where the product rule cannot
be used such as
•
Multiply radical expressions with more than one
term.
•
•
•
Use polynomial special products to multiply radicals
•
•
•
2x ⋅ 3 4x
(
)
Ask students to multiply: 6 x x − 3 . Which previously learned multiplication
technique did you need to solve the problem? If students are unsure, give them the problem
6 x( x − 3) to simplify. Now go back to the original problem.
Give a worksheet with several multiplication problems (FOIL). Ask students to write down
the previously learned multiplication technique they needed to solve.
As a review ask students to simplify: ( A + B ) , ( A − B ) , ( A + B )( A − B )
Give students a radical example of each pattern type to simplify.
As a change of pace, ask students to summarize how old techniques (like factoring,
exponents, and pattern multiplication) are now being applied to these more difficult
problems.
2
2
Add and Subtract Radical Expressions
Overview of Objectives, students should be able to:
1. Add and subtract like radical expressions
2. Add and subtract radical expressions that require
simplification
Objectives:
Main Overarching Questions:
1.
How do you know when to add or subtract radicals?
2.
How do you know when a radical expression is fully simplified?
Activities and Questions to ask students:
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•
Add and subtract like radical expressions
•
•
•
•
•
•
•
•
•
•
•
•
Students have learned this concept in MAT 0024, but as before a quick review discussion will be
useful.
Ask students how they would simplify 2 x + 3 x = 5 x .
Where did the “5” come from?
Why didn’t x become x2?
If students have difficulty, ask “What is 2 apples plus 3 apples?” Is it 5 apples (x) , or 5 tangerines
(x2)?
Now give simple radical example: What is 2 5 + 3 5 . Ask students to consider 5 is “x” or
the apple as in the previous examples.
Write down the process you used to add the radicals. Give several more simple examples
9 + 16 ? Can you simplify like you did before? Why or why not?
If students have trouble and guess 9 + 16 = 25 = 5 ask them if there was another way to
find the answer. 9 + 16 = 3 + 4 = 7 . Why did the first way not work?
What about
What if variables are included? What is 2 2 x + 3 2 x . If students have trouble, phrase it this
way: “What is 2 square root of 2x’s plus 3 square root of 2x’s.” If they say five, ask “We have five
of what?” Remind them of what did not change in the original example.
How can you tell that square roots cannot be added or subtracted together? Write down several
examples where they can and where they cannot be added or subtracted.
Can the same rule hold for cube roots and higher roots? Give students several problems to try
with the process on (with constants and variables) such as 9 3 7 − 4 3 7 ,
6 5 − 3 x + 2 7 + 5 3 x , and
•
Add and subtract radical expressions that require
simplification
9 x − 18 + x − 2
•
•
If 4 + 9 cannot be combined together, is there another way to simplify?
Have students work several examples of this type. What process can you use to simplify?
(simplify and then add or subtract)
•
•
What about 2 12 + 3 ? How can you simplify each radical first? Have you done this before?
What rule or process can you use to help you simplify? How does the “2” on the outside of the
radical change the process?
Write down the process you used to solve the problem.
•
How about 5 3 x + 3 8 x ? Write down the process you used to solve.
•
Give a worksheet with several problems of varying difficulty (with different roots and variables) such as:
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3
•
•
•
54 xy 3 + y 3 128 x and 4 3 x 4 y 2 + 5 x 3 xy 2
Ask students to talk through their strategy on how to solve.
Is there more than one way to perform the examples?
Is it correct to say there are many ways to simplify? Why?
Divide Radical Expressions
Overview of Objectives, students should be able to:
Main Overarching Questions:
1.
1. Use the quotient rule to simplify radical expressions
2. Use the quotient rule to divide radical expressions.
Objectives:
•
Use the quotient rule to simplify radical expressions
How do you divide radicals?
2. How do you know when a radical expression is fully simplified?
Activities and Questions to ask students:
•
Ask students to evaluate:
16
4
16
. Do you see a relationship?
4
•
Ask students to evaluate:
•
Have students work several pairs of examples of this type. Does a pattern exist?
•
What is
•
Do you notice a pattern?
•
Have students draw the conclusion that:
a2
b2
? What about
a2
?
b2
a
=
b
a
b
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policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.
•
•
Use the quotient rule to divide radical expressions
Ask students to show that the rule can be extended to:
n
a
=
b
n
a
n
b
•
Give students a few examples to demonstrate the quotient rule such as:
•
Ask students to rewrite the quotient rule for radicals:
n
a
=
b
n
a
n
b
3
50 x 8
and
27 y 12
•
How could you use the rule to divide two radicals? Write the process you would use.
•
Ask students to divide:
•
40
10
3
Ask students to divide:
13 y 7
x 12
. Write the process you used to simplify.
250 x 5 y 3
3
4
2x3
. Which previously learned technique did you need to solve
the problem?
Rationalize Denominators
Overview of Objectives, students should be able to:
1. Rationalize denominators with one term
a. Square root denominators
b. Higher index root denominators
2. Rationalize denominators with more than one term
Objectives:
Main Overarching Questions:
1. How do you rationalize the denominator of radical expression?
2.
How do you know when a radical expression is fully simplified?
Activities and Questions to ask students:
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•
Rationalize denominators with one term
o Square root denominators
o Higher index root denominators
•
Ask students to simplify:
this type.
•
Ask students to divide:
5 ⋅ 5 = 5 . Do you notice a pattern? Give several more examples of
4
2
. What is different about this problem vs. other division
=
5
5
•
•
problems?
How could you write an equivalent fraction that has no radical in the denominator? Ask students
to think about how they can write equivalent fractions when adding and subtraction fractions.
Write the process and operation you would use.
What could you multiply by to achieve your goal? Write down your process.
Tell students this process is called “rationalizing the denominator”
•
Give an example: rationalize the denominator of
•
•
2x 2
3x
. Write down your steps.
What about for higher order roots? Ask students how they would rationalize:
2
3
4
. Most of
them will assume they only need to multiply by the 3 4 as they did in the previous process.
However this will not work because the product gives us a radical term that cannot be
completely simplified down. Remind them that the goal of rationalizing the denominator is to
remove the radical from the denominator. Ask them to figure out what they could multiply by to
“get rid” of the radical. If they have trouble, ask them to figure out what numbers they can take
the cube root of nicely, i.e.
3
1 = 1 , 3 8 = 2 , etc. Ask them how they would transform
3
4 to
8 through multiplication. What do we need to multiply by?
Have them complete the example and write down a general process to rationalize denominators
with higher order roots. Give a couple of examples for the students to work with higher order
3
•
roots such as:
•
Rationalize denominators with more than one term
10
5
16 x 2
5
,
4
x2 y7
•
Ask students to simplify: (2 + 5 )(2 − 5 ) . Do you notice a pattern? Give several more
examples of this type. What happens to the radical terms?
•
How could you use this pattern to rationalize the denominator of:
2
2+ 5
. What could you
multiply by to remove the radical term? Write the process you would use.
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•
Tell students that if an expression is a + b then its conjugate is a − b or vice versa.
•
Give an example: rationalize the denominator of
•
•
What is the conjugate of the denominator?
Write down your steps to solve.
4
x −1
.
Rational Exponents
Overview of Objectives, students should be able to:
1. Use the definition of a
2. Use the definition of a
1
n
m
−m
Objectives:
Use the definition of a
1
1.
How do you simplify expressions with rational exponents?
2. How can you use rational exponents to simplify radical expressions?
n
3. Use the definition of a n
4. Simplify expressions with rational exponents
5. Simplify radical expressions using rational exponents
•
Main Overarching Questions:
n
3. How do you know when your expression is fully simplified?
Activities and Questions to ask students:
•
1
To motivate the discussion, ask students to consider 8 3 . We want to figure out what this
1
•
•
quantity is, so let’s set a variable equal to this unknown quantity: x = 8 3 .
Although students have not solved equations like this before, ask them to think about what
they could do to both sides of the equation to “get rid of” the fractional power.
Some might consider multiplication or division, but remind them to think about their
exponent rules. It won’t be obvious to cube both sides, so if necessary give a simple
example.
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3
•
1
Once students are on board to cube both sides ask students to simplify: x =  8 3 


3
1
Now, x = 8 = 8 . With this is mind, ask students what they would observe x to be (2).
•
Remind them of the original goal, we wanted to know what x = 8 3 was. We now have
•
3
1
1
2 = 8 3 . So what does the fractional power do? Ask students to then generalize what
•
•
1
1
means. Have students draw the conclusion that a n = n a .
Later on we will be simplifying these rational exponential expressions, so remind students to
consider all the exponent rules they have learned. It might be beneficial to have students list
out all the exponential rules they remember.
Ask students to rewrite the following expressions as radicals and simplify where
a
n
possible: (100)
•
Use the definition of a
m
n
•
2
, (−81)
1
4
, (xy )
1
5
Give some additional problems to go the other way by rewriting as a fractional power:
8
•
1
10
4,
1
to
2
3x y
Ask students to consider what a
difference between a
m
1
n
and a
m
m
n
n
is, keeping in mind that a
1
n
= n a . What is the
? What operation would you need to do to change a
n
. Some will say multiply by m, but remind them we are dealing with exponents. What
do we have to do to multiply powers?
a
•
n
Once students realize we need raise a
of the equation: a
1
n
= n a ,  a

1
m
n
1
n
 =

to the mth power, ask them to do this to both sides
( a)
n
m
m
,  a n  =


( a)
n
m
. Tell students that we can
interchange the power on the outside of the radical and move it to the inside:
 a m n  =


m
 a n  =


•
( a ). Have students draw to the conclusion that  a
( a ).
n
m
n
m
m
n
 =

( a)
n
m
and
Ask students if they see a pattern or an easier way to remember what part of the fractional
power represents the root and what part represents the exponent. Have them generalize
the rule.
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2
•
Give students the example 1000 3 and ask them to rewrite this as a radical expression.
Since there are two ways to rewrite these expressions, how do you know which way is
better?
•
Give students another example like 2 2 and ask them to rewrite as a radical expression.
Which of the two ways is better? Can you generalize when to use one way vs. the other?
Give a few more arithmetic problems to work like the two above, additionally have students
practice going back and forth between radicals and fractional exponents. Make sure to give
•
3
some where the entire expression contains the fractional power, like ( xy ) 7 and some where
4
only parts of the expression are under the power like 2xy
•
Use the definition of a
−m
n
Simplify expressions with rational exponents
7
•
This rule will be much easier to derive. Ask students to recall what a − n is.
•
How could we apply this rule to find a
•
•
4
•
•
•
•
•
Have students conclude that a
−m
n
−m
n
1
=
a
m
?
.
n
Does a negative exponent mean we will get a negative number?
Have students work problems similar to the last concept but with negative exponents.
At this point, we will be taking more complicated rational exponent problems and
simplifying, so if this hasn’t been done, ask students to write down all exponential properties
(rules) they remember. Have them summarize the properties for the class. Remind them
that rational exponents will use all these properties.
Ask students to also consider when an expression is completely simplified.
Have students work several problems of varying degree such as 5
2
3
1
⋅5 3 , x
3
4
1
⋅x 4,
y
y
1
4
5
3
4
,
8
3
 y − 3 4  6 ,  2 x 14  , and  x 13 y −3 2  4 . For each problem, have students summarize which






•
Simplify radical expressions using rational exponents
•
•
exponential properties they used and the ordered in which they simplified. Is there more
than one way to simplify?
This last section will apply all the properties of this section to simplify radical expressions.
Ask students how they might simplify 10 x 5 using rational exponents. Remind them to put
the final answer back in terms of a radical if a fractional power remains. Ask students if they
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•
could have simplified the expression without rational exponents.
Allow students to practice this concept by giving several examples to work out. In these
problems, it’s important to write out (with a phrase) what property they are using at each
step and to keep their work in order. Have them compare not only final answers, but the
order and properties they used. If someone gets the same answer, does it mean he or she
completed the problem in the exact manner you did? Hopefully students will see there are
many ways to simplify these expressions.
Radical Equations
Overview of Objectives, students should be able to:
Main Overarching Questions:
1. How do you solve radical equations?
1. Solve radical equations by using the squaring
property of equality
2. Solve radical equations by using the squaring
property of equality twice.
2. How do you check your solutions are correct?
3.
What is an extraneous solution?
Objectives:
Activities and Questions to ask students:
3. Solve radical equations by using the squaring
property of equality
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Give students a simple example: if x = 2. What is x2 = ? What about if x = 3, x2 = ?
Do you see a pattern? What “operation” are we performing on both sides of the equation?
Have students draw the conclusion that if a = b then a 2 = b 2 (squaring principle)
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Now ask students how they would solve: x = 4 . If students just observe the answer is 16, ask
them how they would solve the equation using the squaring principle. Does it match the solution
you observed?
How can you check your solution is correct? Write down the process you used to solve and
check your answer.
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How do you solve x = −4 ? What happens if you use the squaring principle? How could you
check that 16 is not the solution? Mention that solutions that do not work in an equation but
that are the result of an algebraic method are called “extraneous” solutions
Give another example similar to this one. Do you see a pattern? How could you predict there
would be no solution?
The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the
policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.
•
Solve radical equations by using the squaring
property of equality twice.
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How would you solve: x + 4 − 6 = −2 . How is this example different than the last one? How
would you need to modify your process to solve it? If students take exception to the -2 on the
right hand side, ask them if the radical has been isolated. Remind them that in the previous
simple example, the radical was by itself when we realized the negative on the right hand side
would give an extraneous solution.
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Ask students how they would find:
x + 4 − 2 . Have you done this before? Which
previously learned processes or rules are you using?
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How would you solve: x = x + 4 − 2 ? What is different about this example than the last
ones? How can you use the squaring process at the beginning to help you solve?
After using the squaring property you still have a radical in the equation, now what do you do?
How many radicals do you have now? Does it look similar to the first type of radical equations
you solved? Write the process you would use to continue.
Ask students to summarize the process of solving radical equations of the types studied.
Give students a worksheet with several radical equations (1 and 2 radicals, some with real
solutions, and some with no real solution) to complete. Have them use the process they wrote
down.
Is there more than one way to solve? Compare with your classmates.
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Solve higher order radical equations.
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(
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2
Give students a cube root equation like 3 x + 4 = 4 and ask them to think about how they would
solve it. What is different about this equation? Can we square both sides of the equation like in
our previous examples? Why or why not? If we squared both sides of the equation to “remove”
the square root, what do you think we should do to remove a cube root? What about the nth
root?
Give another simple cube root equation, but this time with a negative number on the right hand
side like: 3 2 x + 6 = −2 . Some students should remember that negatives were a sign of
extraneous solutions (i.e. no solution) with the square root. Ask everyone to consider if this will
be an issue with cube roots?
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To break the previous concept down, consider a very simple example like 3 x = −2 . Mentally,
ask students to find a number whose cube root is -2. Can you find the number? What is it?
Have students draw the conclusion that negatives are not an issue with cube roots.
With square roots (an even index of 2) negatives gave an extraneous solution and with cube
roots (an odd index of 3) negatives produced a real solution. Ask the students to give a pattern
The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the
policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.
•
for all even root equations and odd root equations.
Give some practice problems of varying difficulty (one root only, cube roots and higher). Give at
least one problem with a fractional power like: (4 x − 7 )
consider what a fractional power represents.
1
5
− 8 = −4 . Remind students to
Quadratic Equations
Overview of Objectives, students should be able to:
1. Solve quadratic equations
a. Solve quadratic equations using factoring
b. Solve quadratic equations using the square root
property
2. Solve quadratic equations using the quadratic
formula
3. Solve problems using quadratic equations.
Objectives:
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Solve quadratic equations
o Solve quadratic equations using factoring
o Solve quadratic equations using the square root
property
Main Overarching Questions:
1.
How do you solve quadratic equations?
2.
How many solutions can we expect to get?
Activities and Questions to ask students:
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Give students the general quadratic equation ax 2 + bx + c = 0
How do you know an equation is quadratic? What properties does it have?
Give students a simple product such as 4 ⋅ 0 and ask for the result.
Then reverse the order and ask for the result of 0 ⋅ 4 .
Ask the students if they notice any similarities between the two simple expressions.
Why is the final result the same in each case?
What requirement must be met for the product of two numbers to be 0?
Next, give the students the simple equation: a ⋅ b = 0 . What are the possible solutions to this
equation? Is there more than one solution?
Have students establish the zero product property: if the product of two factors is 0, then either
The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the
policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.
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factor is 0.
Give students a very simple quadratic equation to solve like ( x + 2)(2 x + 3) = 0
How would we solve this equation? If necessary, remind students of what has been discussed
so far this discussion.
Next give students a simple quadratic equation: x 2 + 4 x = 0
Ask students how they might attempt to solve the equation.
How can we use the zero product property to aid in solving the equation?
If we need a product to use the property, how can we transform our sum of terms into a
product? If students have trouble, give students an arithmetic example to illustrate the point:
“How would we write 10 as a product?”
Then, have students solve x 2 + 4 x = 0 . What steps did you use to solve the equation?
Give students another equation to solve using a different factoring method like x 2 + 4 x + 3 = 0
What differences did you notice in solving this equation?
Give students one additional equation that does not have 0 on the right hand side:
x 2 + 3 x + 3 = 1 . What additional steps might be necessary? Summarize the process.
In each example, how many solutions did we get? Is there a relationship between the degree of
the equation and the number of solutions?
Square Root Property:
• Give students a very simple quadratic equation to solve like: x 2 = 4 .
• If students only give x = 2 as the solution ask them if there any other solutions. If necessary,
ask them how many solutions we normally have when solving quadratic equation.
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Next, ask students (this has already been discussed) what x 2 is.
How can we “get rid of” a square on a variable?
Now, going back to x 2 = 4 ask the students what other process we might use to solve the
equation.
If we take the square root of both sides, what additional steps do we need to get both
solutions?
Have students summarize the process of solving quadratic equations by taking square roots.
Give students another quadratic equation to solve: 2 x 2 = 18 . What is different about this
equation? What step could we use to make the equation look similar to the first one?
The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the
policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.
•
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Have students summarize the steps to solve this equation.
What steps would be necessary to solve: 5 x 2 + 2 = 27 ? What is different about this
equation? What step or steps would be necessary to make it look like the first equation?
Finally, have students summarize the steps to solve ( x + 2 ) − 4 = 5 . What is different
about the “squared” portion of this equation?
• After taking the square root of both sides what additional step or steps is necessary?
• Give students the general quadratic equation and quadratic formula: ax 2 + bx + c = 0 and
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Solve quadratic equations using the quadratic
formula
x=
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1.
Solve problems using quadratic equations.
2
2.
3.
4.
5.
6.
− b ± b 2 − 4ac
.
2a
What do you notice about the formula? What does the formula give us? How many solutions
should we get? In what instances would we get 1 solution? No solutions?
Ask students how could they identify what the a, b, and c values are.
Give students the example 3 x 2 + 4 x + 1 = 0 . Ask students to summarize the process they use
to solve the equation. Ask in particular that they summarize the steps in simplifying the
expression.
What about 2 x 2 = 4 x + 1 ? What are the a, b, and c values? What additional step do we need
before plugging into the formula?
Have students revisit the projectile problem. Again, since we haven’t introduced the graph of a
quadratic equation, give the students a graph of the height of projectile vs. time.
Although they don’t have the tools to calculate the maximum height, ask students where they
think the maximum height would occur.
Have students describe at what point on the graph the ball would hit the ground? What is the
height at this point?
Give students a quadratic equations that represents the height of the projectile such as
h = −t 2 + 5t − 6 . If we are asked to figure out when the projectile hits the ground, what value
should we let h be? How can we use what we have learned to figure out the missing time
values?
Once students have solved the equation, ask them how many solutions they found. Are both
solutions valid? Why or why not? Which solution is valid?
The contents of this website were developed under Congressionally-directed grant (P116Z080078) from the U.S. Department of Education. However, those contents do not necessarily represent the
policy of the U.S. Department of Education, and you should not assume endorsement by the Federal Government.