Problem 1P

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.

Additional TEKS (4)(G)

TEKS FOCUSFormulate – create with careful effort and purpose. You can formulate a plan or strategy to solve a problem.

Strategy – a plan or method for solving a problem

Reasonableness – 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.

VOCABULARY

Solving a square root equation may require that you square each side of the equation. This can introduce extraneous solutions.

ESSENTIAL UNDERSTANDING

Solving a Square Root Equation

What is the solution of 3 + 12x − 3 = 8?

3 + 12x - 3 = 8

12x - 3 = 5 Isolate the radical expression.

(12x - 3)2 = 52 Square each side.

2x - 3 = 25

2x = 28 Add 3 to each side.

x = 14 Divide each side by 2.

Check

3 + 12x - 3 = 8 Write the original equation.

3 + 12(14) - 3≟ 8 Substitute 14 for x.

3 + 125≟ 8 Simplify.

3 + 5≟ 8

8 = 8 ✔

6-5 Solving Square Root Equations

C

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.

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Problem 3bl 3

Problem 2

Checking for Extraneous Solutions

What is the solution of 1x + 7 − 5 = x? Check your results.

1x + 7 - 5 = x

1x + 7 = x + 5 Isolate the radical.

11x + 7 22 = (x + 5)2 Square each side.

x + 7 = x2 + 10x + 25 Simplify.

0 = x2 + 9x + 18 Combine like terms.

0 = (x + 3)(x + 6) Factor.

x = -3 or x = -6 Zero-Product Property

Check

1x + 7 - 5 = x

1-3 + 7 - 5 ≟ -3

14 - 5≟ -3

2 - 5≟ -3

-3 = -3 ✔

1x + 7 - 5 = x

1-6 + 7 - 5 ≟ -6

11 - 5≟ -6

1 - 5≟ -6

-4 ≠ -6

The only solution is -3. false

TEKS Process Standard (1)(F)

Solving an Equation With Two Radicals

What is the solution of 12x + 1 − 1x = 1?

12x + 1 - 1x = 1

12x + 1 = 1x + 1 Isolate the more complicated radical.

112x + 122 = 11x + 122 Square each side.

2x + 1 = x + 2 1x + 1

x = 2 1x Isolate 21x.

x2 = 12 1x22 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

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How do you square a binomial?Use the formula, (a + b)2 =a2 + 2ab + b2.

WWhich radical expression should you isolate first?Isolate the more complicated radical first, 22x + 1.

254 Lesson 6-5 Solving Square Root Equations

Problem 4P bl 4

continuedProblem 3

Check

12x + 1 - 1x = 1

12(0) + 1 - 10 ≟ 1

11 - 0≟ 1

1 - 0≟ 1

1 = 1 ✔

12x + 1 - 1x = 1

12(4) + 1 - 14 ≟ 1

19 - 14≟ 1

3 - 2≟ 1

1 = 1 ✔

The solutions are 0 and 4.

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.

Formulate a Plan Write a square root equation that models the data to find a solution to the problem.

The equation to find the side length: s = 2A

The equation to find the perimeter: P = 4s

The equations combined: P = 42A

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.

Determine a Solution

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.

TEKS Process Standard (1)(B)

Area = 72.25 ft²

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m

TT

T

T

Ao

D

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.

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continuedProblem 4

Justify the Solution

P = 4s Use the equation for perimeter based on side length.

34 = 4s Substitute the perimeter calculated above.

344 = s Solve for s.

8.5 = s

s = 2A Use the equation for side length based on area.

8.5 = 2A Substitute the computed side length.

(8.5)2 = (2A)2 Square both sides to solve for area.

72.25 = A The solution checks.

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.

PRACTICE and APPLICATION EXERCISES

ONLINE

HO

M E W O RK

For additional support whencompleting your homework, go to PearsonTEXAS.com.

Solve. Check for extraneous solutions.

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.

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?

s√32

s

s

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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.

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.

√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

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TEXAS Test PracticeT

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 + 62

D. 3 - i

32. Which expression represents the solution of the equation xy =c

a + b solved for a?

F. cb -xy

G. yc

a + b

H. ycx + b

J. yc - xb

x

33. To rationalize the denominator of 54

254 , by what number would you multiply

the numerator and denominator of the fraction?

22. Devise a plan to find the value of x.

x = 52 + 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

30. Apply Mathematics (1)(A) The equation P = 2p5x

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?

258 Lesson 6-5 Solving Square Root Equations