quadratic equations §5.5. objectives by the end of today, you should be able to… solve quadratic...

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.



Step 2) Factor.





Step 2) Factor.

???

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


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



thousandth!

x = 0.438 x = 4.562


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



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 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.


Solve by factoring.


Solve by factoring.

4x 2 250


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.


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.


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

HOMEWORK

• p.266 (7-19)

quadratic equations §5.5. objectives by the end of today, you should be able to… solve quadratic...

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