solve quadratic equations

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Quadratic Equations

• Solving by
• Square rooting
• factoring

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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
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• The degree tells us how many roots a polynomial
  has
• A polynomial of degree n will have n roots.
• The roots may be real and/or complex
• Real roots may be repeated, this is multiplicity

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Methods Used to Solve Quadratic Equations
1. Square Root Property
2. Factoring
3.
Quadratic Formula
4. Graphing
5. Completing the Square
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Why so many methods?
- Some methods will not work for all
equations.
- Some equations are much easier to solve
using a particular method.
- Variety is the spice of life.
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Square Root Method
Basic Steps, you try to get the squared variable
by itself, then take the square root of both
sides and solve
The best time to use it is there is only a
variable squared or a quantity squared.
i.e. x2 25 or (x 2)2
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It is important to remember that there is a
positive and negative square root of every
number.
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Square Root Method

• Example 1
• x2 3 6
• x2 9
• x 3
• x3
• x 3
• x

Example 3 (x 4)2 100 x 4 10i x
4 10i Which is both x 4 10i x 4
10i
Example 2 (x 3)2 25 x 3 5
x 3 5 x 3 5 x 2 x 8
2 complex solutions
2 real solutions
2 real solutions
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Square Root method

• Example 4
• (x 3)2 9
• x 3 3
• x 3 3
• x 6
• x 3 3
• x 0
• x 6, 0

Example 5 9x2 36 0 9x2
36 x2 4 x 2i x 2i,
-2i
Example 6 2g2 32 g2 16
g 4
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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

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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 factor two or
more numbers or expressions have in common
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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
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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

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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
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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 )

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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

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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
• (Denominator numerator)

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