solve by factoring

1
Quadratic Equations

• An Introduction

2
Solving Quadratic Equations

• Degree of an equation with one variable is
  highest exponent or power
• A quadratic equation has no exponents larger than
  2, ax2 bx c 0, where a ? 0.

a cannot equal zeroit must have a squared term
What if b 0 ?
What if c 0 ?
What if b c 0 ?
2x2 4x 0
2x2 4 0
2x2 0
2x2 4x 2 0
The solution to an equation (also called the
roots or zeros) is all of the numbers that make
it true
3
Solve quadratic by Factoring

• Basic steps 1. make the equation equal zero,
• 2. factor the non zero
  side
• 3. set each factor zero
  and solve
• is best used when the quadratic equation
• is easily factorable
• It is important to remember that it is based on
  zero product property
• and not all quadratics can be factored

4
Solve quadratic by factoring
Factoring writing a number or expression as a
product of numbers and/or expressions Greatest
Common Factor (GCF) is the largest factors two or
more numbers or expressions have in common
5
Zero Product Property
When the product of 2 or more factors is zero,
then at least one of the factors must equal zero

• If A B 0 then
• A 0, or B 0,
• or
• both A and B equal 0.

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Solving by Factoring

• Example 1
• 3x2 12x 0
• 3x(x 4) 0
• 3x 0 x 4 0
• x 0 x 4

Example 2 16x2 4x 0 4x(4x 1) 0 4x 0
or 4x 1 0 x 0 -1 -1
4x -1 x -1/4
1. Make it equal to zero 2. Factor the non
zero side 3. Set each factor equal to zero and
solve each
7
Solving by Factoring
Example 4 6x2 15x 6x2 15x 0 3x(2x
5) 0 3x 0 or 2x 5 0 x 0
-5 -5 2x -5
x -5/2

• Example 3
• 2x2 8x
• 2x2 8x 0
• 2x(x 4) 0
• 2x 0 x 4 0
• x 0 x 4

1. Make it equal to zero 2. Factor the non
zero side 3. Set each factor equal to zero and
solve each
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Solving quadratic, 1 variable Equations

• So, what about this one?
• 
  
• 
  
• 
  

You may feel like you should distribute
But if you do, it does not help you solve it
So lets start again. Notice that we already have
2 factors that equal zero. This is already
factored. We can use the zero product property
Separate the factors and make each equal To zero.
Now I have two equations that I can Solve. The
1st one is already done.

These are both possible solutions, check to If
they both make the original equation true

• 1. Make it equal to zero
• 2. Factor the non zero side
• Set each factor equal to zero
• and solve each

9
Solve quadratic by factoring

• Steps to factoring a quadratic trinomial equation
• ax2 bx c 0 where a 1.
• Make it equal to zero
• Find two numbers the multiply to get
• c, but also add to get b
• 3. Write the factors, fill in the numbers
• with the correct signs
• (x )(x )

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Solving by Factoring

• Example 1
• p2 17p 72
• p2 17p 72 0
• (p )(p ) 0
• ( p 9 )(p 8 ) 0
• p 9 0 or p 8 0
• p 9 or p 8
• 9, 8

Example 2 x2 6x 7 x2 6x 7 0 (x
)(x ) 0 ( x 7 )(x 1 ) 0 x 7
0 or x 1 0 x 7 or x 1
-7, 1
11
Solve quadratic by factoring

• Steps to factoring a quadratic trinomial equation
• ax2 bx c 0 where a gt 1.
• Make it equal to zero
• Factor out the GCF, if there is one
• If a is now 1 go on,
  if a is still gt 1 skip to step
  6
• Find two numbers the multiply to get
• c, but also add to get b
• 5. Write the factors, fill in the numbers
• GCF (x )(x )

12
Solve quadratic by factoring

• Steps to solving a quadratic trinomial equation
  by factoring where ax2 bx c 0 where a gt 1
• Draw and X, put ac on top
• and b on the bottom
• Find 2 numbers that multiply to get
• the top and add to get the bottom
• and write them on the sides
• Divide each side number by a, leave as a fraction
• Write each factor starting at the bottom
• GCF(denominatorx numerator)( denominatorx
  numerator) 0
• 10. Separate each factor and make to zero
  and solve each

13
Factor the polynomial 3x2 5x 2 0
6

• 1.Factor out GCF
• 2. Multiply and put at top of he X
• 3. Put at the bottom of X
• 4. Find the pair of factors of 6 that add to 5
• 5. Divide each of the factors by
• 6. Write the factors
• (DenominatorX numerator)

2
1
3
1
3
3
5
Factors of 6 Sum of factors
1 6
7
2 3
5
-1 -6
-7
-2 -3
-5