Download 3.1 Solving Equations Using Addition and Subtraction

9-3A Solving Quadratic Equations
by Finding Square Roots.
Algebra 1
Glencoe McGraw-Hill
Linda Stamper
You must learn how to use a calculator! There are many
makes and models. Read the instruction booklet.
Enter a problem into the calculator for which you
already know the answer. For example: 4
2nd √ 4
=
2
Keystrokes for
TI-30X IIS
4
√
=
2
Keystrokes for
TI-30X A
Evaluate the expression. Give the exact value, if possible.
Otherwise, approximate to the nearest hundredth. You
may use a calculator for this section.
Example 1
8  2.83
What is the positive square root of 8?
Example 2  11  3.32
What is the negative square root of 11?
Example 3  27  5.20
What is the positive and negative square root of 27?
hundredth
s
Inverse Operations
Recall the use of inverse operations to solve equations.
What is the inverse operation of addition?
What is the inverse operation of subtraction?
What is the inverse operation of multiplication?
What is the inverse operation of division?
What is the inverse operation of a square number?
The inverse of a square number is a square root.
A quadratic equation is an equation that can be written
in the standard form: ax2  bx  c  0
where a  0. a is called the leading coefficient.
Quadratic equations can have one solution, two solutions
or no real solutions.
If b = 0, the equation becomes ax2  c  0
One way to solve a quadratic equation of this form is to
isolate the x2 on one side of the equation. Then find the
square root(s) of each side.
Remember: Squaring a number
and finding the square root(s)
of a number are inverse
operations.
Solve the equation. Write the solutions as integers if
possible. Otherwise, round to the nearest tenth.
x2  4  0
How can you tell this is a
4 4
quadratic equation?
2
x
 4
Isolate the square term.
 x2   4
Undo the square by using square root.
 x  2
Evaluate the radicals.
x  2 and  x  2
x  2
Do not give
this
One of the equations
is not solved for a
answer!An equation is not
positive variable!
considered solved if the variable is
negative. What do you get when you undo
the negative variable?
Solve the equation. Write the solutions as integers if
possible. Otherwise, round to the nearest tenth.
Isolate the square term.
Undo the square by using square root.
Evaluate the radicals.
Remember the variable
cannot have a + sign
because a negative
variable is not solved.
n2  5  0
5 5
n2  5
 n2   5
n 5
n  2.2
Solve the equations. Write the solutions as integers if
possible. Otherwise, round to the nearest tenth.
Example 4
x2  16  0
 16  16
x2  16
 x2   16
x  4
Example 5
m2  11  0
 11  11
m2  11
 m2   11
m   11
m  3.3
Remember the variable
cannot have a + sign
because a negative
variable is not solved.
Example 6
k 2  100
 k2   100
k  10
Solve the equations. Write the solutions as integers if
possible. Otherwise, round to the nearest tenth.
Example 7
3x2  11  11
Example 8
6x2  150  0
Example 9
Example 10
27  3y2  0
x2  15  0
Solve the equations. Write the solutions as integers if
possible. Otherwise, round to the nearest tenth.
Example 7
3x2  11  11
 11  11
3x2  0
3
3
x2  0
 x2  0
x0
Example 8
6x2  150  0
 150  150
6x2  150
6
6
x2  25
 x2   25
x  5
Solve the equations. Write the solutions as integers if
possible. Otherwise, round to the nearest tenth.
Example 9
Example 10
x2  15  0
27  3y2  0
 15  15
 27
 27
2
2

3y
  27
x   15
3
3
2
 x    15
y2  9
 15 is undefined
 y2   9
no real solution
y  3
Remember the variable
cannot have a + sign
because a negative
variable is not solved.
Solve the equations. Write the solutions as integers if
possible. Otherwise, round to the nearest tenth.
Remember a ± sign indicates two solutions (roots).
Factor the P.S.T.
x2  18x  81  8
x  92  8
 x  92   8
x9   8
x  9  8
There are two
solutions (roots):
x  9  8
x  9  8
 6.2,  11.8
Solve the equations. Write the solutions as integers if
possible. Otherwise, round to the nearest tenth.
Example 11
Example 12
x  4 2  49
y  32 
Example 14
Example 15
Example 16
4x2  36  0
x2  10x  25  7
x2  4x  4  5
Hint: Isolate
the squared
term on one side
of the equal sign.
5
Example 13
b2  14b  49  36
Solve the equations. Write the solutions as integers if
possible. Otherwise, round to the nearest tenth.
Example 11
Example 12
y  32 
2
x  4   49
 x  42   49
x47
x  4  7
x   4  7 or x   4  7
3,  11

y  32
5
 5
y 3   5
y 3 5
y  3  5 or y  3  5
5.2, 0.8
Solve the equations. Write the solutions as integers if
possible. Otherwise, round to the nearest tenth.
Example 13
b2  14b  49  36
b  7 2  36
 b  7 2   36
b7  6
b76
b  7  6 or b  7  6
13, 1
Example 14
4x2  36  0
4x2  36
 4x2   36
2x   6
2x  6 or 2x  6
6
6
x
or x 
2
2
x  3
Solve the equations. Write the solutions as integers if
possible. Otherwise, round to the nearest tenth.
Example 15
x2  10x  25  7
2
x  5  7
2
 x  5   7
x 5   7
x 5 7
x  5  7 or x  5  7
7.6, 2.4
Example 16
x2  4x  4  5
x  22  5
 x  22   5
x 2   5
x  2  5
x  2  5 or x  2  5
0.2,  4.2
9-A5 Handout A5.