Solving 2nd Degree Equations
Second degree equations equations are equations that have an x^{2} term as its
highest power. They
are also called Quadratic equations.
Zero Factor Rule
The zero product rule says
if you have two terms whose product (factors) equal zero then
either the first factor is equal to zero
or the second factor is equal to zero
This rule helps us solve equations that have products (factors) that are
equal
to zero.
(x − 3)(x + 4) = 0
(x + 2)(x − 2) = 0
(4x)(x − 6) = 0
x(5x + 2) = 0
If the polynomial is a second degree equation then the polynomial will have
exactly two factors. If
the product of the two factors equals zero you can use the Zero Product Rule to
find the two
numbers that are solutions to the second degree equation.
Solving 2nd Degree Equations
Step 1: Be sure that the problem has an equation with two factors whose product
is zero.
Step 2: Set each factor with an x term equal to zero. This will give you two
separate equations.
Step 3: Solve each equation separately. The 2 numbers are both solutions to the
original equation.
Step 4: You can check the 2 solutions by substituting either of the numbers into
the original equation
and checking to see that they make original make the equation true.
Solve each equation. 
Example 1 
Example 2 
(x −5)(x + 6) = 0 
(4x)(x − 2) = 0 
then 
then 
x −5 = 0 or x + 6 = 0 
4x = 0 or x − 2 = 0 
and solving each
equation for x gives 
and solving each
equation for x gives 
x = 5 or x = −6 
x = 0 or x = 2 
Check x = 5 
Check x = −6 
Check x = 0 
Check x = 2 
(5 −5)(5 + 6) = 0
(0)(11) = 0 
(−6 − 5)(−6+ 6 ) = 0
(−6 −5)(−6 + 6) = 0 
(4 • 0)(0 − 2) = 0
(0)(−2) = 0 
(4 • 2)(2 − 2) = 0(8)(0) = 0 
Both x = 5 and x = 6 are
solutions to 
Both x = 0 and x = 2 are
solutions to 
(x − 5)(x + 6) = 0 
(4x)(x −2) = 0 
Standard Form of a Second Degree Equation
The standard form of a second degree equation requires the x^{2} term to
be written first and be
positive. The x term is written second and the constant term is written last.
That expression is set
equal to 0.
Standard Form Examples
Solving a Second Degree Equation
If we factor a quadratic equation in standard form we get two factors that have
a product of zero.
If we then Use the Zero Factor rule we can solve each of
these equations.
Both numbers are solutions to the original second degree
equation.
To Solve a Second Degree Equation for x:
Step 1: Get the terms in Standard Form and set equal to zero.
Step 2. Factor (Factor out the GCF, The Difference of 2 Perfect Squares, Easy
Trinomials)
Step 3. Set each factor that has an x term equal to zero.
Step 4. Solve each equation for x. Remember that second degree equations have 2
solutions.
Example 1

Example 2 
Example 3 
Solve 6x^{2} −12x = 0 
Solve 10x^{2} + 5x = 0 
Solve x^{2} −9 = 0 
6x(x − 2) = 0 
5x (2x +1) = 0 
(x − 3) (x + 3) = 0 
Set each factor = to 0 
Set each factor = to 0 
Set each factor = to 0 
6x = 0 x −2 = 0 
5x = 0 2x +1 = 0 
x − 3 = 0 x + 3 = 0 
Solve each equation for x 
Solve each equation for x 
Solve each equation for x 
x = 0 or x = 2 
x = 0 or

x = 3 or x = −3 
Both numbers are solutions to the original second degree
equation.
Example 4

Example 5 
Example 6 
Solve 4x^{2} −25 = 0 
Solve 8x^{2} − 32 = 0 
Solve x^{2} + 8x +15 = 0 
(2x −5) (2x + 5) = 0 
8(x^{2} − 4) = 0
8(x −2) (x + 2) = 0 
(x + 5) (x + 3) = 0 
Set each factor = to 0 
Set each factor with an x term =
to 0 
Set each factor = to 0 
2x −5 = 0 2x + 5 = 0 
x − 2 = 0 x + 2 = 0 
x + 5 = 0 x + 3 = 0 
Solve each equation for x 
Solve each equation for x 
Solve each equation for x 

x = 2 or x = −2 
x = −5 or x = −3 
Example 7

Example 8 
Example 9 
Solve x^{2} −6x − 7 = 0 
Solve 6x^{2} + 7x − 3 = 0 
Solve 3x^{2} − 2x − 4 = 0 
(x − 7) (x +1) = 0 
(3x −1) (2x + 3) = 0 
(x −2) (3x + 4) = 0 
Set each factor = to 0 
Set each factor = to 0 
Set each factor = to 0 
x − 7 = 0 x +1 = 0 
3x −1 = 0 2x + 3 = 0 
x −2 = 0 3x + 4 = 0 
Solve each equation for x 
Solve each equation for x 
Solve each equation for x 
x = 7 or x = −1 

x = 2 or 
Example 10

Example 11 
Example 12 
Solve 8x^{2} = 4 x 
Solve 16x^{2} = 81 
Solve x^{2} = −9x −14 
Put in standard form 
Put in standard form 
Put in standard form 
8x^{2} − 4 x = 0 
16x^{2} − 81= 0 
x^{2} + 9x +14 = 0 
factor 8x^{2} − 4 x = 0 
factor 16x^{2} −81 = 0 
factor x^{2} + 9x +14 = 0 
4x(2x −1) = 0 
(4x −9) (4 x + 9) = 0 
(x + 7) (x + 2) = 0 
Set each factor = to 0 
Set each factor = to 0 
Set each factor = to 0 
4x = 0 2x −1= 0 
4x −9 = 0 4x + 9 = 0 
x + 7 = 0 x + 2 = 0 
Solve each equation for x 
Solve each equation for x 
Solve each equation for x 
x = 0 or x =1/2 

x = −7 or x = −2 
Example 7

Example 8 
Example 9 
Solve 8x^{2} = 4 x 
Solve 4x^{2} = 25 
Solve x^{2} = −9x −14 
Put in standard form 
Put in standard form 
Put in standard form 
8x^{2} − 4 x = 0 
4x^{2} −25 = 0 
x^{2} + 9x +14 = 0 
factor 8x^{2} − 4 x = 0 
factor 4x^{2} −25 = 0 
factor x^{2} + 9x +14 = 0 
4x(2x −1) = 0 
(2x −5) (2x + 5) = 0 
(x + 7) (x + 2) = 0 
Set each factor = to 0 
Set each factor = to 0 
Set each factor = to 0 
4x = 0 2x −1= 0 
2x −5 = 0 2x + 5 = 0 
x + 7 = 0 x + 2 = 0 
Solve each equation for x 
Solve each equation for x 
Solve each equation for x 
x = 0 or x =1/2 
x = 5/2 or x = −5/2 
x = −7 or x = −2 
