1. Solving Quadratic Equations
Zero Product Property
Using Factoring to solve Quadratic Equations

2. Degree of an equation
Equations with one variable, the degree is equal to the highest exponent
3x + 4 = first degree equation
second degree equation,
also called quadratic equations

3. Solving a First Degree Equation
You have had a good amount of experience with this:
3x + 4 = to solve, get the variable alone
subtract 4 on both sides
3x = divide both sides by 3
x = variable alone, coefficient of 1
3(3) + 4 = is the only number that makes
= this equation true

4. Solving a Second Degree Equation
Getting the variable alone, with a coefficient of one will work in some 2nd degree equations.
subtract 2 on both sides
divide both sides by 4
take the square root of both sides
2 or -2 can make the equation true

5. Solving a 2nd Degree Equation
What about this one?
Or this one?
We need new strategy…

6. Lets go over some vocabulary 2nd degree equations—we are going to call them quadratic equations or quadratics

7. Lets go over some vocabulary Factoring a number or expression— means to break it down into two or more parts that are multiplied together.

8. Zero Product Property
If A • 5 = 0 then
A = 0

9. Zero Product Property
If • B = 0 then
B = 0

10. Zero Product Property If A • B = 0 then A = 0 or B = 0,
0 • B = or A • 0 = 0 or
both A and B equal 0
0 • 0 = 0

11. Solve (x + 3)(x - 5) = 0 If A • B = 0
We can use the Zero Product Property whenever we have two factors that equal zero
we know that either
A = or B = 0
x + 3 = or x - 5 = 0
Solve each equation.
x = or x = 5

12. Solve (2a + 4)(a + 7) = 0 A • B = 0 So…. 2a + 4 = 0 or a + 7 = 0
{-2, -7}

13. A(B) = 0 So…….. t = 0 or t - 3 = 0 or t = 3 {0, 3}
Solve t(t - 3) = 0
A(B) = 0
So……..
t = or t - 3 = 0
or t = 3
{0, 3}

14. Solve (y – 3)(2y + 6) = 0
{-3, 3}
{-3, 6}
{3, 6}
{3, -6}

15. Solving a Quadratic Equation
What about this one?
Or this one?
Lets take them one at a time

16. Solving a Quadratic Equation
What about this one?
if we are going to use the
zero product property,
it needs to equal zero
It also has to be the
product of 2 factors
We have to factor it
Find GCF, then divide by it

17. Solving a Quadratic Equation
What about this one?
if we are going to use the
zero product property,
it needs to equal zero
Now break up the two
factors and make each
equal to zero
Then solve each

18. Solve x2 - 11x = 0 GCF = x x(x - 11) = 0 x = 0 or x - 11 = 0
{0, 11}

19. Solve a2 - 24a +144 = 0 a2 - 24a + 144 = 0 (a - 12)(a - 12) = 0
This one does not have a GCF other than 1.
We use the X box to factor and solve this one.
a2 - 24a = 0
(a - 12)(a - 12) = 0
a - 12 = 0
a = 12
{12}
Put in descending order
Squared term has to be positive
Put 1st and 3rd term in the box
We have two boxes and 1 term left, the xgame will show us how to split them up.
Multiply the 1st and 3rd term and put on top of x
The middle term goes on the bottom
The numbers you find go in the two remaining boxes
Find the GCF of each column and row for your factors a
-12
144a2 a
-12a a2
-12a
-12a
-12
-12a
144
-24a

20. Solve x2 + 2x = 15
This one does not have a GCF other than 1.
We use the X box to factor and solve this one.
x2 + 2x – 15 = 0
x2 + 2x – 15 = 0
(x - 3)(x + 5) = 0
x – 3 = x + 5 = 0
x = 3 or x = -5
{3, -5}
Put in descending order
Squared term has to be positive
Put 1st and 3rd term in the box
We have two boxes and 1 term left, the xgame will show us how to split them up.
Multiply the 1st and 3rd term and put on top of x
The middle term goes on the bottom
The numbers you find go in the two remaining boxes
Find the GCF of each column and row for your factors x +5
-15x2 x x2 5x 5x
-3x -3
-3x
-15 2x

21. 4 steps for solving a quadratic equation by factoring
Set the equation equal to 0.
2 terms, factor by distribution or
or difference of two squares
4 terms, reverse box
3 terms, X box
No matter how many terms, always start by finding the GCF