8-7 solving quadratic equations by using square roots · 2018-03-06 · holt mcdougal algebra 1 8-7...

Holt McDougal Algebra 1

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

Solving Quadratic Equationsby Using Square Roots

Holt Algebra 1

Warm Up

Lesson Presentation

Lesson Quiz



8-7Solving Quadratic Equations by Using Square Roots

Warm UpFind each square root.

Solve each equation.

5. –6x = –60 6.

7. 2x – 40 = 0 8. 5x = 3

6 11

–25

1. 2.

3. 4.

x = 10 x = 80

x = 20



Solve quadratic equations by using square roots.

Objective



Some quadratic equations cannot be easily solved by factoring. Square roots can be used to solve some of these quadratic equations. Recall from Lesson 1-5 that every positive real number has two square roots, one positive and one negative.



Negative

Square root of 9

When you take the square root of a positive number and the sign of the square root is not indicated, you must find both the positive and negative square root. This is indicated by ±√

Positive and negative

Square roots of 9

Positive

Square root of 9



The expression ±3 is read “plus or minus three”

Reading Math



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


x2 = 121

x = ± 11

The solutions are 11 and –11.



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✓



Check It Out! Example 1b


x2 = 0

x = 0

The solution is 0.



square roots.

Substitute 0 into the

original equation.

Check x2 = 0

(0)2 0

0 0✓



x2 = –16


square is negative.


Check It Out! Example 1c




If necessary, use inverse operations to isolate the squared part of a quadratic equation before taking the square root of both sides.



Example 2A: Using Square Roots to Solve Quadratic

Equations


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


16x2 – 49 = 0

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

Divide both sides by 16.


sides. Use ± to show both

square roots.



Example 2B Continued


Check 16x2 – 49 = 0

49 – 49 0

16x2 – 49 = 0

49 – 49 0 ✓ ✓

.




Solve by using square roots. Check your answer.

100x2 + 49 = 0

100x2 + 49 = 0–49 –49

100x2 =–49


There is no real number

whose square is negative.







(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.



Check It Out! Example 2b Continued


Check (x – 5)2 = 16

(9 – 5)2 16

42 16

16 16

(x – 5)2 = 16

(1 – 5)2 16

(– 4)2 16

16 16✓ ✓



When solving quadratic equations by using square roots, you may need to find the square root of a number that is not a perfect square. In this case, the answer is an irrational number. You can approximate the solutions.



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.


sides.

Evaluate on a calculator.x 5.48



Example 3B Continued


–3x2 + 90 = 0


Check Use a graphing calculator to support your answer.

Use the zero function. The approximate solutions are 5.48 and – 5.48.





0 = 90 – x2

+ x2 + x2 0 = 90 – x2

x2 = 90

Add x2 to both sides.


sides.






2x2 – 64 = 0

2x2 – 64 = 0

+ 64 + 64

x2 = 32


Add 64 to both sides.

Divide by 2 on both sides.


sides.



Check It Out! Example 3c


x2 + 45 = 0

x2 + 45 = 0– 45 – 45

x2 = –45


square is negative.





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


Evaluate on a calculator.

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.




Check It Out! Example 4

A house is on a lot that is shaped like a trapezoid. The solid lines show the boundaries, where xrepresents 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.


sides.

Evaluate on a

calculator.

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 1

Solve using square roots. Check your answers.

1. x2 – 195 = 1

2. 4x2 – 18 = –9

3. (x + 7)2 = 81

4. Solve 0 = –5x2 + 225. Round to the nearest hundredth.

± 14

± 6.71

– 16, 2



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

8-7 solving quadratic equations by using square roots · 2018-03-06 · holt mcdougal algebra 1 8-7...

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