Section 7.1

Extraction of Roots and Properties of Square Roots

7.1 Lecture Guide: Extraction of Roots and Properties of Square Roots

Objective 1: Solve quadratic equations by extraction of roots.

A quadratic equation is an equation that can be written in the _________________________ form .The quadratic equation can be rewritten as . The two solutions of this equation are the two numbers whose square is 3. We can denote these solutions by One solution is and the other is . The notation “ ” is read plus or minus. This process of solving quadratic equations is called extraction of roots.

2 0ax bx c

3.x 3x 3x

2 3 0x 2 3x

Extraction of Roots Example

If k is a positive real number, then the equation has two real

solutions, :

.

2x kx k

andx k x k

2 3x

3 3x or x

Use extraction of roots to determine the exact solutions of each quadratic equation.

1. 2.2 16x 2 7x

3. 4.22 18x 24 12 0x

Use extraction of roots to determine the exact solutions of each quadratic equation.

Objective 2: Use the product rule for radicals to simplify square roots.

Product Rule for

Square Roots

Algebraic Example

Numerical Example

Ifare both real numbers, then

andx y

.xy x y

12 4 3

4 3

2 3

Simplify each square root using the product rule for square roots.

5. 4 3

Simplify each square root using the product rule for square roots.

6. 40

Simplify each square root using the product rule for square roots.

7. 75

Simplify each square root using the product rule for square roots.

8. 32

Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.

9.2 24x

Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.

10. 2 54x

Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.

11. 22 64x

Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.

12. 25 100 0x

Objective 3: Use the Quotient rule for radicals to simplify square roots.

Quotient Rule for Square Roots

Algebraic Example

Numerical Example

If are both real numbers and , then .

andx y

0y x xy y

25 2536 36

56

Simplify each square root using the quotient rule for square roots.

13.425

Simplify each square root using the quotient rule for square roots.

14. 1116

Simplify each square root using the quotient rule for square roots.

15. 75

12

Simplify each square root using the quotient rule for square roots.

16. 45

20

Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.

17. 225 11x

Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.

18. 24 3 0x

The process of rewriting a radical expression so that the _________________ does not have any radicals in it is called rationalizing the denominator. This process uses

the fact that for ,

Objective 4: Simplify expressions of the form by rationalizing the denominator.

a

b

0x 2 .x x x x

Simplify each expression by rationalizing the denominator.

19.3

2

Simplify each expression by rationalizing the denominator.

20. 5

6

Simplify each expression by rationalizing the denominator.

21. 57

Simplify each expression by rationalizing the denominator.

22.1220

Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.

23. 25 9x

Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.

24. 210 15 0x

In the following box we extend the method of extraction of roots to a quadratic equation that contains the square of a binomial on the left side of the equation.

Extraction of Roots Example

For a positive real number k and nonzero real numbers a

and b: If , then

.

or 2ax b k

ax b k

22 1 3x

2 1 3x 2 1 3x

Solve each quadratic equation using extraction of roots.

25. 21 4x

Solve each quadratic equation using extraction of roots.

26. 23 25x

Solve each quadratic equation using extraction of roots.

27. 22 1 5 41x

Solve each quadratic equation using extraction of roots.

28. 23 2 6 10x

Solve each equation.

29. 22 5 9x

Solve each equation.

30. 2 5 9x

31. Interest Rate A certificate of deposit earns an annual rate of interest r that is compounded twice a year. An investment of $1,300 grows to $1,412.85 by the end of 1 year. Use the

formula to approximate the annual interest

rate to the nearest tenth of a percent.

2

12r

A P

32.Checking a Solution You can check solutions of any equation by substitution or by graphing.

(a) In problem 18 from earlier in this section you were asked to solve The solutions of this equation are

Check these solutions of this by substitution.

24 3 0.x 3

.2

x

3,3,1 by 5,5,1

(b) Check those same solutions by graphing and using the trace feature.

21 4 3Y x