Download 6-5 Solving Square Root Equations

6-5
Solving Square Root Equations
TEKS FOCUS
VOCABULARY
TEKS (4)(F) Solve quadratic and square root equations.
TEKS (1)(B) Use a problem-solving model that
incorporates analyzing given information, formulating
a plan or strategy, determining a solution, justifying the
solution, and evaluating the problem-solving process
and the reasonableness of the solution.
ĚFormulate – create with careful effort and purpose. You
can formulate a plan or strategy to solve a problem.
ĚStrategy – a plan or method for solving a problem
ĚReasonableness – the quality of being within the realm of
common sense or sound reasoning. The reasonableness of
a solution is whether or not the solution makes sense.
Additional TEKS (4)(G)
ESSENTIAL UNDERSTANDING
Solving a square root equation may require that you square each side of the equation.
This can introduce extraneous solutions.
Problem 1
P
Solving a Square Root Equation
What is the solution of 3 + 12x − 3 = 8?
3 + 12x - 3 = 8
12x - 3 = 5
Do you need to
introduce a t sign
here?
No, when you take the
square root of each
side of an equation you
do; but here you are
squaring both sides of
the equation.
(12x - 3)2 = 52
Isolate the radical expression.
Square each side.
2x - 3 = 25
2x = 28
x = 14
Add 3 to each side.
Divide each side by 2.
Check
C
3 + 12x - 3 = 8
3 + 12(14) - 3 ≟ 8
3 + 125 ≟ 8
Write the original equation.
Substitute 14 for x.
Simplify.
3 + 5≟8
8=8 ✔
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Problem 2
TEKS Process Standard (1)(F)
Checking for Extraneous Solutions
What is the solution of 1x + 7 − 5 = x? Check your results.
1x + 7 - 5 = x
How do you square
a binomial?
Use the formula,
(a + b)2 =
a2 + 2ab + b2.
1x + 7 = x + 5
Isolate the radical.
11x + 7 2 2 = (x + 5)2
Square each side.
x + 7 = x2 + 10x + 25
0=
x2
+ 9x + 18
Simplify.
Combine like terms.
0 = (x + 3)(x + 6)
Factor.
x = -3 or x = -6
Zero-Product Property
Check
1x + 7 - 5 = x
1x + 7 - 5 = x
1-3 + 7 - 5 ≟ -3
1-6 + 7 - 5 ≟ -6
14 - 5 ≟ -3
11 - 5 ≟ -6
2 - 5 ≟ -3
1 - 5 ≟ -6
-3 = -3 ✔
-4 ≠ -6
The only solution is -3.
false
Problem
bl
3
Solving an Equation With Two Radicals
Which radical
expression should
you isolate first?
Isolate the more
complicated radical first,
22x + 1.
What is the solution of 12x + 1 − 1x = 1?
W
12x + 1 - 1x = 1
12x + 1 = 1x + 1
112x +
122
= 11x +
122
Isolate the more complicated radical.
Square each side.
2x + 1 = x + 2 1x + 1
x = 2 1x
x2 = 1 2 1x 2 2
Isolate 21x.
Square each side.
x2 = 4x
x2 - 4x = 0
Subtract 4x from each side.
x(x - 4) = 0
Factor.
x = 0 or x = 4
Zero-Product Property
continued on next page ▶
254
Lesson 6-5
Solving Square Root Equations
Problem 3
continued
Check
12x + 1 - 1x = 1
12x + 1 - 1x = 1
12(0) + 1 - 10 ≟ 1
12(4) + 1 - 14 ≟ 1
11 - 0 ≟ 1
19 - 14 ≟ 1
1 - 0≟1
3 - 2≟1
1=1 ✔
1=1 ✔
The solutions are 0 and 4.
Problem
P
bl
4
TEKS Process Standard (1)(B)
Using a Problem-Solving Model
A gardener is making square planting beds with different
areas. Based on the area of the planting bed, the gardener
wants to know the perimeter of the planting bed so that
she knows how much fencing is needed to enclose the
planting bed. How much fencing will she need to enclose the
planting bed? Explain the problem-solving model you use to
determine if your answer is reasonable.
Analyze Given Information The square root of the area gives
the side length of the planting bed. The perimeter is four times
the side length.
How can you use area
to find perimeter?
You know how to find
side length based on the
area of a square, and
you know how to find
perimeter based on side
length. Combine these
two equations into a third
equation.
Area = 72.25 ft²
Formulate a Plan Write a square root equation that
models the data to find a solution to the problem.
m
The equation to find the side length:
T
s = 2A
The equation to find the perimeter:
T
P = 4s
T equations combined:
The
P = 4 2A
A represents the area of the planting bed, s represents the length of one side
of the planting bed, and P represents the perimeter of the planting bed.
o
Determine a Solution
D
P = 42A
Use the equation for perimeter based on area.
= 42(72.25)
Substitute the known value for area.
= 4(8.5)
Use the positive square root.
= 34
Multiply.
The gardener needs 34 feet of fencing to fence the perimeter of the square
planting bed.
continued on next page ▶
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Problem 4
continued
Justify the Solution
P = 4s
Use the equation for perimeter based on side length.
34 = 4s
Substitute the perimeter calculated above.
34
4 =s
Solve for s.
8.5 = s
s = 2A
8.5 = 2A
( )
Use the equation for side length based on area.
Substitute the computed side length.
(8.5)2 = 2A 2
Square both sides to solve for area.
72.25 = A
The solution checks.
NLINE
HO
ME
RK
O
Evaluate the Reasonableness of the Solution You know that 82 = 64 and
92 = 81. Since the area of the planting bed, 72.25 square feet, is between 64
and 81, it makes sense that the side length is between 8 and 9.
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Solve. Check for extraneous solutions.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1. 13x + 7 = x - 1
2. 1-3x - 5 = x + 3
3. 111x + 3 - 2x = 0
4. 13x + 13 - 5 = x
5. 1x + 7 + 5 = x
6. 1x + 7 - x = 1
7. Use a Problem-Solving Model (1)(B) A vegetable tray in the shape of a regular
hexagon has an area of 450 cm2. What is the length of each side of the hexagon? Use
a problem-solving model by analyzing the given information, formulating a plan or
strategy, determining a solution, justifying the solution, and evaluating the
reasonableness of the solution.
s
s
s√3
2
8. Apply Mathematics (1)(A) A stop sign is a regular octagon, formed by cutting
triangles off the corners of a square. If a stop sign measures 36 in. from top to
bottom, what is the length of each side of the octagon?
256
Lesson 6-5
Solving Square Root Equations
Solve. Check for extraneous solutions.
9. 13x = 1x + 6
10. 13x + 2 - 12x + 7 = 0
11. 15 - x - 1x = 1
12. 13x + 1 - 1x + 1 = 2
13. 12x + 6 - 1x - 1 = 2
14. 13 - x + 1x + 2 = 3
15. What is the solution? 1x + 11 = 4
16. You can find the area A of a square whose side is s units with the formula A = s2.
What is the best estimate for the side of a square with an area of 32 m2 ?
A. 4.2 m
C. 8.0 m
B. 5.7 m
D. 16 m
17. Explain Mathematical Ideas (1)(G) A student said that 4 and 1 are the
solutions of the problem shown. Describe and correct the student’s error.
√x + 2 = x
√x = x - 2
(√x)2 = (x - 2)2
x = x2 - 4x + 4
0 = x2 - 5x + 4
0 = (x - 4)(x - 1)
STEM
18. Apply Mathematics (1)(A) The velocity v of an object dropped from a tall
building is given by the formula v = 164d, where d is the distance the object
has dropped. Solve the formula for d.
19. Write an equation that has two radical expressions and no real roots.
20. Analyze Mathematical Relationships (1)(F) You have solved equations containing
square roots by squaring each side. You were using the property that if a = b
then a2 = b2. Show that the following statements are not true for all real numbers.
a. If a2 = b2 then a = b.
b. If a … b then a2 … b2.
21. A teacher asked students why it is necessary to check for extraneous roots when
squaring both sides of the equation. Which of the following answers is the best?
Is this answer complete? Explain.
A. Because the squared equation can have negative solutions.
B. Because squaring is multiplication, and any multiplication is a potential
source of extraneous solutions.
C. Because when you square both sides of the equation a = b, you add to
the solution set the solutions of the equation a = -b.
D. Because any operation with an equation may result in extraneous solutions.
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22. Devise a plan to find the value of x.
x = 5 2 + 22 + 12 + g
For each set of values, determine which is greater without using a calculator.
23. 16 or 12 + 1
24. 13 + 111 or 5
25. 110 or 12 + 13
26. 119 + 13 or 15 + 113
Solve. Check for extraneous solutions.
27. 22x + 3 = x
28. 2x + 6 - 4 = x
29. 2x = 1 + 24 - 8x
x
30. Apply Mathematics (1)(A) The equation P = 2p 5 9.8
gives the period of
a pendulum in seconds, where x is the length of the pendulum in meters. A
scientist needs to build a pendulum with a period that is 1 second longer than
the period of a 5.2-meter pendulum. To the nearest thousandth of a meter, how
long should the new pendulum be?
TEXAS Test Practice
T
31. A problem on a test asked students to solve a fifth-degree polynomial
equation with rational coefficients. Adam found the following roots: -11.5, 12,
2i + 6
2 , -12, and 3 - i. His teacher wrote that four of these roots are correct, and
one is incorrect. Which root is incorrect?
A. -11.5
B. 12
C.
2i + 6
2
D. 3 - i
32. Which expression represents the solution of the equation xy = a +c b solved
for a?
c
x
F. b - y
yc
G. a + b
yc
H. x + b
J.
yc - xb
x
4 4 , by what number would you multiply
33. To rationalize the denominator of 5
25
the numerator and denominator of the fraction?
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Lesson 6-5
Solving Square Root Equations