QUADRATIC EQUATIONS§5.5

OBJECTIVES

By the end of today, you should be able to…

• Solve quadratic equations by factoring and graphing.

What does it mean to solve an equation?

Quick Review:When you solve an equation, you’re finding the values that make the statement true.

INVESTIGATION

Solve the following equations:

QUADRATIC EQUATIONS

The standard form of a quadratic equation is ax2 + bx + c = 0, where a ≠ 0.

You can solve some quadratic equations in standard form by factoring the quadratic expression and then using…

The Zero-Product Property!

THE ZERO-PRODUCT PROPERTY

The Zero-Product Property helps us solve quadratic equations that have been factored.

The Zero-Product Property states:

If ab = 0, then a = 0 or b = 0.

For example, if (x + 3)(x - 7)=0, then (x + 3) = 0 or (x - 7) = 0.

EXAMPLE 1: SOLVING BY FACTORING

Solve by factoring.Step 1) Write in standard form.

Step 2) Factor.

Step 3) Use the Zero-Product Property.

Step 4) Solve for x.

EXAMPLE 2: SOLVING BY FACTORING

Solve by factoring.Step 1) Write in standard form.

Step 2) Factor.

Step 3) Use the Zero-Product Property.

Step 4) Solve for x.

EXAMPLE 3: SOLVING BY FACTORING

Solve by factoring.Step 1) Write in standard form.

Step 2) Factor.

Step 3) Use the Zero-Product Property.

Step 4) Solve for x.

EXAMPLE 4: SOLVING BY FACTORING

Solve by factoring.Step 1) Write in standard form.

Step 2) Factor.

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SOLVING BY GRAPHING

You can solve a quadratic equation in standard form by graphing its related quadratic function,

y = ax2 + bx +c.

Quadratic equation cannot

be solved by factoring

Quadratic equation can be solved by graphing!

SOLVING BY GRAPHING

Equation Solutions Vertex X-intercept Y-intercept

x2 – x – 6

x2 + 8x + 7

x2 – 6x + 9

x2 - 16

x2 + 4x – 5

SOLVING BY GRAPHING

If we want to know what values of x make ax2 + bx +c = 0, where should we look?

We want to find where y = 0!

• When the graph of the function intersects the x-axis, the value of the function is zero, and each x-value is a zero of the function.

• A zero of a function is a solution of the equation ax2 + bx +c = 0.

A.K.A: Find the

x-intercepts!

EXAMPLE 4: SOLVING BY GRAPHING

Solve by graphing.Step 1) Graph the related function, y = x2 + 6x + 4.

Step 2) Use the CALC feature to find the two zeros of the function.

Round to the nearest

thousandth!

x = -5.236 x = -0.764

EXAMPLE 5: SOLVING BY GRAPHING

Solve by graphing.Step 1) Graph the related function, y = x2 – 5x + 2.

Step 2) Use the CALC feature to find the two zeros of the function.

Round to the nearest

thousandth!

x = 0.438 x = 4.562

EXAMPLE 6: SOLVING BY GRAPHING

Solve by graphing.Step 1) Graph the related function, y = -x2 – 2x + 7 or y = x2 + 2x – 7.

Step 2) Use the CALC feature to find the two zeros of the function.

Round to the nearest

thousandth!

x = 1.828x = -3.828

QUADRATIC EQUATIONSPT. 2

§5.5

OBJECTIVES

By the end of today, you should be able to…

• Simplify square roots.

• Solve quadratic equations by finding square roots.

SOLVE.What do you remember?...

Solve by factoring: (Pick 4)

x2 – 6x – 7 = 0x2 + 6 = 7x9x2 – 8x = 1

x2 + 5x + 4x2 + 6 = 5x2x2 – x = 6

Solve:

X2 – 12 = 025x2 – 9 = 0

SOLVING QUADRATIC EQUATIONS

If the quadratic equation is missing a middle term, or a b term, we can solve the equation in a

different way.

We can solve the equation by taking the square root of

each side.

Why?

EXAMPLE 1: SIMPLIFYING RADICALS

EXAMPLE 2: SIMPLIFYING RADICALS

EXAMPLE 3: SIMPLIFYING RADICALS

EXAMPLE 4: SOLVING BY FINDING SQUARE ROOTS

Solve by factoring.

5x 2 1800

Step 1) Rewrite in form ax2 = c.

Step 2) Isolate x2.

Step 3) Simplify.

Step 4) Take the square root of each side.

EXAMPLE 5: SOLVING BY FINDING SQUARE ROOTS

Solve by factoring.

EXAMPLE 6: SOLVING BY FINDING SQUARE ROOTS

Solve by factoring.

4x 2 250

EXAMPLE 6: SOLVING BY FINDING SQUARE ROOTS

Solve by factoring.

EXAMPLE 7: REAL WORLD CONNECTION-FIREFIGHTINGSmoke jumpers are in free fall from the time they jump out of a plane until they open their parachutes. The function y = -16t2 + 1600 models a jumpers height y in feet at t seconds for a jump from 1600 ft. How long is a jumper in free fall if the parachute opens at 1000 ft?

Step 1) Substitute 1000 for y.

Step 2) Isolate t2.

Step 3) Take the square root of each side.

EXAMPLE 8: FALLING OBJECTSThe function y = -16x2 + 270 models the height y in feet of Wile E. Coyote x seconds after he runs off the edge of a cliff that is 270 feet tall. How long does it take Wile E. Coyote to hit the ground?

Step 1) Substitute 0 for y.

Step 2) Isolate x2.

Step 3) Take the square root of each side.

CHALLENGENow it’s time for you to be the teacher! Write a quadratic equation with the given solutions.

HOMEWORK

• p.266 (7-19)