4.5 Solve Quadratic Equations by Finding Square Roots

Square root

A number r is a square root of a number s if r² = s.

Radical

An expression of the form where s is a number or expression

Radicand

The number s beneath the radical sign

Rationalizing the denominator

The process of eliminating a radical from the denominator of a fraction

Conjugates

The expressions a + and a - , used to rationalize the denominator, whose product is always a rational number

ROPERTIES OF SQUARE ROOTS (a > 0, b> 0)

	Example
Product Property	= _____ · ______
Quotient Property	=

Example 1

a. = _____ · ______ = _____

b. = _____ = _____ · ______ = ______

c. = = _{_____}

Example 2

Rationalize denominators of fractions

Simplify (a) and (b)

a.

3

= · =

b. = ·

=

=

Example 3

Solve a quadratic equation

Solve (y - 6)² = 8.

(y - 6)² = 8

Original equation

_(y - 6)² _ = __32___

Multiply each side by __4__.

_(y - 6)_ = _±____

Take square roots of each side.

y = _6_±_____

Add ___6___ to each side.

y = _6 ±______

Simplify.

The solutions are _6 + ____ and _6 - _____.

Example 4

Model a dropped object with a quadratic function

Water Balloon A water balloon is dropped from a window 59 feet above the sidewalk. How long does it take for the water balloon to hit the sidewalk?

Solution

h = -16t^2 + h0	Write height function.
_0_ = -16t^2 + _59_	Substitute _0_ for h and _59_ for h0.
_-59___ = -16t^2	Subtract _59_ from each side.
_____ = t^2	Divide each side by _-16_.
________ = t	Take square roots of each side.
_±1.9__ » t	Use a calculator.

Reject the negative solution, _-1.9_, because time must be positive. The water balloon will fall for about _1.9__ seconds before it hits the ground.