Holt McDougal Algebra 1 8-7 Solving Quadratic Equations by Using Square Roots 8-7 Solving Quadratic...

Holt McDougal Algebra 1

8-7 Solving Quadratic Equations by Using Square Roots8-7 Solving Quadratic Equations

by Using Square Roots

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8-7 Solving Quadratic Equations by Using Square Roots

Solve quadratic equations by using square roots.

Objective



Example 1A: Using Square Roots to Solve x2 = a

Solve using square roots. Check your answer.

x2 = 169

x = ± 13

The solutions are 13 and –13.

Solve for x by taking the square root of both sides. Use ± to show both square roots.

Substitute 13 and –13 into the original

equation.

x2 = 169 (–13)2 169 169 169

Check x2 = 169 (13)2 169 169 169



Example 1B: Using Square Roots to Solve x2 = a

Solve using square roots.

x2 = –49

There is no real number whose square is negative.

There is no real solution.



Check It Out! Example 1a

Solve using square roots. Check your answer.

x2 = 121

x = ± 11

The solutions are 11 and –11.

Solve for x by taking the square root of both sides. Use ± to show both square roots.

Substitute 11 and –11 into the original

equation.

x2 = 121 (–11)2 121

121 121

Check x2 = 121 (11)2 121 121 121



Example 2A: Using Square Roots to Solve Quadratic Equations

Solve using square roots.

x2 + 7 = 7

–7 –7 x2 + 7 = 7

x2 = 0

The solution is 0.

Subtract 7 from both sides.

Take the square root of both sides.



Example 2B: Using Square Roots to Solve Quadratic Equations

Solve using square roots. 16x2 – 49 = 0

16x2 – 49 = 0+49 +49 Add 49 to both sides.

Divide both sides by 16.

Take the square root of both sides. Use ± to show both square roots.



Check It Out! Example 2b

Solve by using square roots. Check your answer.

(x – 5)2 = 16

Take the square root of both sides. Use ± to show both square roots.

(x – 5)2 = 16

x – 5 = ±4

Write two equations, using both the positive and negative square roots, and solve each equation.

x – 5 = 4 or x – 5 = –4

+ 5 + 5 + 5 + 5

x = 9 or x = 1

The solutions are 9 and 1.



Example 3A: Approximating Solutions

Solve. Round to the nearest hundredth.

x2 = 15


Evaluate on a calculator.

The approximate solutions are 3.87 and –3.87.

x 3.87



Example 3B: Approximating Solutions


–3x2 + 90 = 0–3x2 + 90 = 0

–90 –90

x2 = 30



Divide by – 3 on both sides.


Evaluate on a calculator.x 5.48



Check It Out! Example 3a

Solve. Round to the nearest hundredth.0 = 90 – x2

+ x2 + x2 0 = 90 – x2

x2 = 90

Add x2 to both sides.





Check It Out! Example 3b


2x2 – 64 = 0

2x2 – 64 = 0+ 64 + 64

x2 = 32


Add 64 to both sides.

Divide by 2 on both sides.




Check It Out! Example 3c


x2 + 45 = 0

x2 + 45 = 0– 45 – 45 x2 = –45

There is no real number whose square is negative.

There is no real solution.




Example 4: Application

Ms. Pirzada is building a retaining wall along one of the long sides of her rectangular garden. The garden is twice as long as it is wide. It also has an area of 578 square feet. What will be the length of the retaining wall? Let x represent the width of the garden.

lw = A Use the formula for area of a rectangle.

Substitute x for w, 2x for l, and 578 for A.

2x x = 578

l = 2w

2x2 = 578

Length is twice the width.



Example 4 Continued

2x2 = 578

x = ± 17



Negative numbers are not reasonable for width, so x = 17 is the only solution that makes sense. Therefore, the length is 2w or 34 feet.

Divide both sides by 2.



Check It Out! Example 4

A house is on a lot that is shaped like a trapezoid. The solid lines show the boundaries, where x represents the width of the front yard. Find the width of the front yard, given that the area is 6000 square feet. Round to the nearest foot.

(Hint: Use )

2x

2x

x

Use the formula for area of a trapezoid.



Check It Out! Example 4

Substitute 2x for h and b1, x for b2 , and 6000 for A.

Divide by 3 on both sides.



Negative numbers are not reasonable for width, so x ≈ 45 is the only solution that makes sense. Therefore, the width of the front yard is about 45 feet.



Lesson Quiz: Part II

5. A community swimming pool is in the shape of a trapezoid. The height of the trapezoid is twice as long as the shorter base and the longer base is twice as long as the height.

The area of the pool is 3675 square feet. What is the length of the longer base? Round to the nearest foot.

(Hint: Use )

108 feet

Holt McDougal Algebra 1 8-7 Solving Quadratic Equations by Using Square Roots 8-7 Solving Quadratic...

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