Lesson on Factoring Quadratic Equations

Factoring Quadratic Equations

​Dr. Tom Giles

​Factoring Remediation

​When factoring, your expression should be alone or set = 0.

​Factoring Remediation

​Always check for a GCF!

• ​GCF: Is the Greatest Common Factor. You are looking for factors the terms have in common.

• Checking for a GCF will make the factoring easier.

• If "c" = 0, you can only factor using the GCF.​

• When factoring the GCF, you are basically undistributing (dividing)​

• If a is negative, always factor out a -1​

Factoring Quadratic Equations

​Factoring Remediation

​Factoring by grouping

​Factoring ax² +bx + c   when a does NOT equal 1

1. Multiply a and c.

2. Find the factors of the product from step 1 that "sum" to b

3. "Bust Up your b" - rewrite the expression with 4 terms by using the factors found in step 2 to write b as two terms instead of 1.

4. Create your 2 groups *BE CAREFUL

5. Factor using GCF's (twice)

6. Write the expression in factored form.​

Multiple Choice

What is ALWAYS the first step when factoring?

• Look for the GCF

• Write a, b, and c

• Difference of squares

• Group

Multiple Choice

What is the GCF of the terms below?

24x³ + 36x²

• 8x

• 6x²

• 12x²

• 4x

• 12x³

Multiple Choice

Which of the following methods should be used to factor the expression $$5x-10$$  ?

• GCF

• Prayer

• AC Method

• Grouping

Multiple Choice

Factor: x² - 5x

• 1(x - 5)

• x(x - 5)

• x(x - 5x)

• x²(1 - 5)

Multiple Choice

Which is a common factor of 12x and 27?

• x

• 12

• 3

• 9

Multiple Choice

Factor: 10x + 15

• 100x + 150

• 1(10x + 15)

• 10(x + 5)

• 5(2x + 3)

Multiple Choice

What is the GCF? 5x²+20x

• 5

• 5x

• x

• 20x

​Factoring Remediation

​Factoring ax² +bx + c   when a = 1

• Step 1: Find the factors of c ( hint: use factor pairs)

• Step 2: If c is positive, both terms are ++ or --. If c is negative, the terms are + AND -. If b is -, both terms are -. If b is +, both terms are +. The biggest factor will always be the sign of b.

• Step 3: Of the factor pairs found in step 1, which combination "adds" to b

• Rewrite the expression in factored form using the factors from step 3.

​Factoring Remediation

​Sign "rules" to help us pick the factors.

​

• If c is positive (ax²+bx+c or ax²-bx+c)​ the factors will have the same sign and you will "add" them together.

• If c is negative ​(ax²+bx-c or ax²-bx-c)​ the factors will have opposite signs and you will "subtract" them.

• The sign of b is always the sign of the larger factor in the factor pair.​

Multiple Choice

Factor: $$x^2+4x+3$$  

When looking for the two integers:

which integer pair combination needs to be met?

• Product is 3

  Sum is 4 

• Product is 4

  Sum is 3

Multiple Choice

To factor $$x^2-8x-20$$   I am looking for a pair of numbers that _________

• have a product of -20 and a sum of -8

• have a product of -8 and a sum of -20

• have a product of 20 and a sum of 8

• have a product of 8 and a sum of 20

Multiple Choice

To factor $$x^2-8x-20$$   I am looking for a pair of numbers that _________

and the largest factor is_________

• are the same sign and both are positive

• are different signs and the larger factor is negative and the smaller factor is positive

• are different signs and the larger factor is positive and the smaller factor is negative

• are the same sign and both are negative

Multiple Choice

Which of the following is a factor of x²-10x+24?

• x-4
• x+4
• x+6
• x-2

Match

Match the following trinomials to their factored form

• $$x^2+10x\ +24$$  

• $$x^2-10x\ +24$$  

• $$x^2-10x\ -24$$  

•   $$x^2+10x\ -24$$  

(x+6)(x+4)

(x-6)(x-4)

(x-12)(x+2)

(x-2)(x+12)

• (x+6)(x+4)

• (x-6)(x-4)

• (x-12)(x+2)

• (x-2)(x+12)

Match

Match the following trinomials to their factored form.

• $$x^2+5x\ +6$$  

• $$x^2+6x\ +\ 5$$  

• $$x^2-x-6$$  

• $$x^2+5x\ -\ 6$$  

• $$x^2-5x+6$$  

(x+2)(x+3)

(x+1)(x+5)

(x-3)(x+2)

(x+6)(x-1)

(x-2)(x-3)

• (x+2)(x+3)

• (x+1)(x+5)

• (x-3)(x+2)

• (x+6)(x-1)

• (x-2)(x-3)

Multiple Choice

Factor
d² - 10d + 25

• (d + 5) (d - 5)
• (d - 15) (d + 10)
• (d + 5) (d - 20)
• (d - 5) (d - 5)

Fill in the Blank

Factor completely:

$$x^2+15x+56$$  

(Do not use spaces in your answer)

The quadratic x²+2x-3=0​

in factored form is (x+3)(x-1) =0

*Set each factor equal to 0 and solve

x+3 = 0 and x-1 = 0 ​

-3 -3 +1 +1

x = -3 x = 1​

​Example

To find the solutions (sometimes called roots or zeros):

• Set ea​ch factor equal to 0

• Solve for x.​

​Solving (finding solutions)

​Factoring Remediation

​The solutions are x = -3 and x = 1

Multiple Choice

What is the next step in solving after getting the quadratic in factored form?

(x + 2)(x - 3) = 0

• Plug in zero for x

• Multiply the binomials

• FOIL

• Set each factor equal to zero

Multiple Choice

What equations would I use to solve x² + 9x - 36 =0

• x + 12 = 0

  and x - 3 = 0

• x + 9 = 0

  and

  x - 4 = 0

• x - 12 = 0

  and

  x + 3 = 0

• x² + 9 = 0

  and

  x - 36 = 0

Multiple Choice

What equations would I use to solve x² + 14x - 32 =0

• x + 16 = 0

  and x - 2 = 0

• x + 7 = 0

  and

  x - 7 = 0

• x - 16 = 0

  and

  x + 2 = 0

• x² + 14 = 0

  and

  x - 32 = 0

Multiple Choice

Convert x² - 9x + 20 = 0 to factored form and solve.

• (x-10)(x+2)=0

  x = -2, 10

• (x+4)(x+5)=0

  x = -5, -4

• (x-4)(x-5)=0

  x = 4, 5

• Not factorable

Multiple Choice

To factor the expression either by slide-divide or by grouping, I need to find the factor pairs for what number?

$$2x^2+5x-12$$  

• -24

• 10

• -12

• -60

Multiple Choice

When factoring by grouping what is the 4-term expression for the quadratic trinomial 3x² + 7x +2

(What is the expression I get when I ''bust up the b" in step 3?)

• 3x² + 2x + 3x + 2

• 3x² + 3x + 4x + 2

• 3x² + x + 6x + 2

• 3x² + 42x -7x + 2

Multiple Choice

The which shows the correct way to group the expression

2x² + 3x - 14x - 21 ?

• $$(2x^2+3x)(-14x-21)$$  

• $$(2x^2+3x)-(14x-21)$$  

• $$(2x^2-14x)-(3x+21)$$  

• $$\left(2x^2+3x\right)-\left(14x\ +\ 21\right)$$  

Multiple Choice

To factor 6x² - 11x + 4 by using the slide-divide method, my first step is to _______

• slide the 6 to the 4 and multiply

• slide the 4 to the 6 and multiply

• slide the 6 to the 4 and divide

• slide the 4 to the 6 and divide

Fill in the Blank

Factor: 6x² - 11x + 4

Multiple Choice

Factor Completely: 
3x² + 18x +15

• 3(x² + 6x + 5)
• (x + 5)(x + 1)
• 3(x + 5)(x + 1)
• Prime

Match

Match the following trinomials to their factored forms

• 2x²+11x+12

• 2x²+5x-12

• 2x²-5x-12

• 2x²-x-6

• 2x²+7x+6

(2x+3)(x+4)

(2x-3)(x+4)

(2x+3)(x-4)

(2x+3)(x-2)

(2x+3)(x+2)

• (2x+3)(x+4)

• (2x-3)(x+4)

• (2x+3)(x-4)

• (2x+3)(x-2)

• (2x+3)(x+2)

Multiple Choice

Solve by factoring: 5x² + 13x + 6 = 0

• x = -2 and x = -3/5
• x = 2 and x = 3/5
• x = -2 and x = 3/5
• x = 2 and x = -3/5

Multiple Choice

Which is the correct factoring and solutions? $$2h^2+11h+5\ =\ 0$$  

• (2h +5)(h + 1)

    h =   $$-\frac{5}{2}$$  ,  $$-1$$  

• (2h +1)(h + 5)

  h =   $$-\frac{1}{2},\ -5$$  

• (2h +2)(h + 3)   

  h =  $$-1,\ -3$$  

• (2h +3)(h + 2) 

   h =   $$-\frac{3}{2},\ -2$$  

Multiple Choice

Solve
6x² + 17x + 12 = 0

• x = -4/3, x = -3/2
• x = 9, x = 8
• x = 4, x = 3
• x = -4, x = -3

Multiple Choice

7x² - 26x - 8 = 0

• x = -1 and x = -7/8
• x = 4 and x = -2/7
• x = -4 and x = 2/7
• x = 4 and x = -7/2

​Factoring Remediation

​Using Graphs to help us Factor.

• The x-intercepts are the solutions to your quadratic equation.

• The x-intercept can be used to write the factors

  an x-intercept of (-3,0) means you have a factor of (x+3) * notice the signs​

• This is not the preferred method to factor. Not all x-intercepts are integers and we cannot always tell exactly what the x-intercepts are.

Multiple Choice

What are the zeros?

• x= 0 and x= -4
• x= 0 and x= 4
• y= 0
• x= 2

Multiple Choice

Which could be a factor of the equation in the graph?

• (x+4)

• (x-4)

• (x-2)

• It is impossible to tell from the graph.

Multiple Select

What are the factors of the quadratic in the graph?

• (x+1)

• (x-1)

• (x+5)

• (x-5)

• (x-4)

Multiple Choice

What are the solutions of the quadratic function in the graph?

• x = -1 and x= 5

• x = 1 and x = -5

• x = -1, x=-2 and x=5

• It is impossible to tell from the graph.

Multiple Choice

What are the x-intercepts?

• (-1,0) and (1,0)

• (0,-1) and (0,1)

• Both intercepts cannot be determined from the graph.

• (-1,0) and (-2,0)

Multiple Choice

Which equation matches the x-intercepts x=5 and x= -7 ?

• (x+5)(x-7)=0

• (x-5)(x+7)=0

• (x+5)(x+7)=0

• (x-5)(x-7)=0

​Factoring Remediation

​Factoring using the Slide-Divide Method

​Factoring ax² +bx + c   when a does NOT equal 1

1. Slide a to c and multiply

2. Rewrite the expression with a =1 and c= product found in step 1.

3. Factor the expression from ​step 2

4. Divide the 2nd term of teach factor by a.

5. Simplify fractions created in step 4.

6. If any ​fractions are left, bottoms up

   (move the denominator to the coefficient of x.​)