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# Solving Quadratic Equations - PowerPoint PPT Presentation

Solving Quadratic Equations. Review of Solving Quadratic Equations. ax 2 +bx +c = 0 A quadratic equation is a 2nd degree equation whose graph is a parabola. Solve with Perfect Square Binomial. Get Perfect Squares on Both Sides of Equation. Take Square Root of Perfect Squares

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ax2 +bx +c = 0

A quadratic equation is a 2nd degree equation whose graph is a parabola.

Solve with Perfect Square Binomial

Get Perfect Squares on Both Sides of Equation.

Take Square Root of Perfect Squares

Solve Positive Number & Neg. Number

Check Answers by Putting into Original Eq.

Solve with Perfect Square Binomial

Get Perfect Squares Binomial.

Take Square Root of Both Sides of Eq

Simplify

Check Answers by Putting into Original Eq.

Set Eq = 0

Factor

Set Each Factor = 0

Check Answers by Putting into Original Eq.

What If You Can’t Factor The Equation?

The “X” factor procedure

doesn’t work which means it

can not be factored

-36

4

Review of a Trinomial Square

(x+5)2 = x2 + 10x + 25

When you square a binomial you get a TRINOMIAL SQUARE

What is Completing The Square?

?

x2 + 10x +

WHAT WILL MAKE A TRINOMIAL SQUARE?

1

-1

x

x2

-x

-x2

?

x2 + 10x +

b

It takes 25 ones to complete this square

Note that 25 is half of b squared

(10/2)2 = 52 = 25

2

b

2

Remember this to Complete The Square

b

This is now a trinomial square

x2+6x+9 is a Trinomial Square

x+3

x+3

x2+6x+9

(x+3)2 =x2 +6x+9

More Examples of

CTS

Half of b Squared

Half of b Squared

More Examples of

CTS

Half of b Squared

More Examples of

CTS

Half of b Squared

-31

4

Solve by Completing The Square

First Put c on One Side of equation.

Solve by Completing The Square

First Put c on One Side of equation.

Divide by a

Skip this step when a=1

Solve by Completing The Square

b

First Put c on One Side of equation.

Divide by a

Complete The Square

(b/2)2=(4/2)2=4

Solve by Completing The Square

First Put c on One Side of equation.

Divide by a

Complete The Square

Write Trinomial Square as a Binomial Square

Solve by Completing The Square

First Put c on One Side of equation.

Divide by a

Complete The Square

Write Trinomial Square as a Binomial Square

Square Root both sides

Solve by Completing The Square

First Put c on One Side of equation.

Divide by a

Complete The Square

Write Trinomial Square as a Binomial Square

Square Root both sides

Solve for x (Note 2 Answers)

Solve by Completing The Square

First Put c on One Side of equation.

Divide equation by a

Complete The Square

(b/2)2=

4. Write Trinomial Square as a Binomial Square

5. Square Root both sides

6. Solve for x (Note 2 Answers)

Solve by Completing The Square

First Put c on One Side of equation.

Divide by a

Complete The Square

(b/2)2=(8/2)2=42=16

4. Write Trinomial Square as a Binomial Square

5. Square Root both sides

6. Solve for x (Note 2 Answers)

Solve by Completing The Square

First Put c on One Side of equation.

Divide by a

Complete The Square

(b/2)2=

4. Write Trinomial Square as a Binomial Square

5. Square Root both sides

6. Solve for x (Note 2 Answers)

Solve by Completing The Square

First Put c on One Side of equation.

Divide by a

Complete The Square

(b/2)2=(2/2)2=1

4. Write Trinomial Square as a Binomial Square

5. Square Root both sides

6. Solve for x (Note 2 Answers)

Solve by Completing The Square

First Put c on One Side of equation.

Divide by a

Complete The Square

(b/2)2=(6/2)2=32=9

4. Write Trinomial Square as a Binomial Square

5. Square Root both sides

6. Solve for x (Note 2 Answers)

Solve by Completing The Square

First Put c on One Side of equation.

Divide by a

Complete The Square

(b/2)2=(9/2)2= 81/4

4. Write Trinomial Square as a Binomial Square

5. Square Root both sides

6. Solve for x (Note 2 Answers)

Solve by Completing The Square

First Put c on One Side of equation.

Divide by a

Complete The Square

(b/2)2=(2/2)2=1

4. Write Trinomial Square as a Binomial Square

5. Square Root both sides

6. Solve for x (Note 2 Answers)

So far we have solved quadratic equations with

FACTORING (Solves Equations with Integer Solutions) & COMPLETING THE SQUARE (Solves All quadratic equations)

2

+

- b

X=

2 a

a=1

b

c

Put Equation into Std. Form (ax2+bx+c=0)

a=1

b

c

Put Equation into Std. Form (ax2+bx+c=0)

Plug a, b & c into the Quad. Eq.

a=1

b

c

Put Equation into Std. Form (ax2+bx+c=0)

Plug a, b & c into the Quad. Eq.

a=1

b

c

Put Equation into Std. Form (ax2+bx+c=0)

Plug a, b & c into the Quad. Eq.

Solve for x

a=1

b

c

Put Equation into Std. Form (ax2+bx+c=0)

a=1

b

c

Put Equation into Std. Form (ax2+bx+c=0)

Plug a, b & c into the Quad. Eq.

a=1

b

c

Put Equation into Std. Form (ax2+bx+c=0)

Plug a, b & c into the Quad. Eq.

a=1

b

c

Put Equation into Std. Form (ax2+bx+c=0)

Plug a, b & c into the Quad. Eq.

Solve for x

a=1

b

c

Put Equation into Std. Form (ax2+bx+c=0)

Plug a, b & c into the Quad. Eq.

Solve for x

Put Equation into Std. Form (ax2+bx+c=0)

Plug a, b & c into the Quad. Eq.

Solve for x

a=1

b

c

Put Equation into Std. Form (ax2+bx+c=0)

Plug a, b & c into the Quad. Eq.

Solve for x

a

b

c

The DiscriminantThe Discriminant equals b2-4ac

The Discriminant is what’s

The DiscriminantThe Discriminant equals b2-4ac

2 Solutions if positive

1 Solution if 0

0 Solutions if negative

The Discriminant tells you how many solutions or roots

y

x

This quadratic has two roots so the

Discriminant must be positive

The Discriminant tells you how many solutions or roots

X

This quadratic has one roots so the

Discriminant must be zero

The Discriminant tells you how many solutions or roots

X

This quadratic has no roots so the

Discriminant must be negative

Find The Number of Solutions (Roots)

Identify a, b & c.

Put values into the discriminant b2-4ac

If Pos. 2 Solutions If Zero 1 Solution If Neg. 0 Solutions

The discriminant is neg. so there are no solutions

Determine The Number of Solutions or Roots

0 Solutions

2 Solutions

1 Solutions

Minus B plus or minus the square root

B squared minus 4AC

All over 2A, That’s the quadratic equation.

It’s the greatest equation of all time.

It even has a wonderful rhyme.

If you don’t know it, It would be a crime

Math is the most taught subject in the world

So you’d better believe me

Bone up to pass the SAT

You won’t end up a weasel.

SOLVING

EQUATIONS

Solutions are Roots

Where Graph crosses

the x-axis

What Method To Use

• SquareRoot when there is a perfect square on one side of eq.

• Use “X” Factor when trinomial is factorable

• Use quadratic formula or complete the square if not factorable