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Solving 2nd Degree Equations

Second degree equations equations are equations that have an x2 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 x2 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 6x2 −12x = 0 Solve 10x2 + 5x = 0 Solve x2 −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 4x2 −25 = 0 Solve 8x2 − 32 = 0 Solve x2 + 8x +15 = 0
(2x −5) (2x + 5) = 0 8(x2 − 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 x2 −6x − 7 = 0 Solve 6x2 + 7x − 3 = 0 Solve 3x2 − 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 8x2 = 4 x Solve 16x2 = 81 Solve x2 = −9x −14
Put in standard form Put in standard form Put in standard form
8x2 − 4 x = 0 16x2 − 81= 0 x2 + 9x +14 = 0
factor 8x2 − 4 x = 0 factor 16x2 −81 = 0 factor x2 + 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 8x2 = 4 x Solve 4x2 = 25 Solve x2 = −9x −14
Put in standard form Put in standard form Put in standard form
8x2 − 4 x = 0 4x2 −25 = 0 x2 + 9x +14 = 0
factor 8x2 − 4 x = 0 factor 4x2 −25 = 0 factor x2 + 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
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