In this algebra lesson, we will discuss how factoring can be used to solve
Quadratic Equations, which are equations of the form:
ax^{2} + bx + c = 0 where a, b and c are numbers and a ≠ 0.
The simplest way to factoring quadratic equations would be to find common factors. Sometimes, the first step is to factor out the greatest common factor before applying other factoring techniques.
In some cases, recognizing some common patterns in the equation will help you
to factorize the quadratic equation. For example, the quadratic equation could be a Perfect Square Trinomial (Square of a Sum or Square of a Difference) or Difference of
Two Squares.
In other cases, you will have to try out different possibilities to get the
right factors for quadratic equations. This is still manageable if the
coefficient of x^{2} is 1. If the
coefficient of x^{2} is greater than 1 then you may want to consider using the Quadratic formula.
Factoring Out Common Factors
We can factorize quadratic equations by looking for values that are common.
Example: 
x^{2} + 3x = 0 

We find that the two terms have x in common. We “take out” x from each term. 

x(x + 3) = 0 

We have two factors when multiplied together gets 0. We know that any number multiplied by 0 gets 0. So, either one or both of the terms are 0 i.e. 

x = 0 or
isolate variable x

This tells us that the quadratic equation
x^{2} + 3
x = 0 can have two values (two solutions) for
x which are
x = 0 or
x = –3
How to solve a quadratic equation by factoring out the greatest common factor?
Examples:
Solve
x
^{2} + 4x = 0
14x
^{2}  35x = 0
Factoring and solving quadratic equations by factoring out the greatest common factor
Example:
12x
^{2} + 18x = 0
4x
^{2} = 20x
How to factor and solve quadratic equations when the first step is to factor out the greatest common factor before apply other factoring techniques?
Example:
Factor
5x
^{2}  45 = 0
3x
^{2}  6x  45 = 0
Factoring Completely
The first step in all factoring problems is to check to see if a Greatest Common Factor can be factored out of the original polynomial.
Example:
Factor
3x
^{2} + 6x  24
Factoring Out The Greatest Common Factor
To find the GCF of a polynomial
1. Write each term in factored form.
2. Identify the factors common to all terms.
3. Factor out the GCF.
Example:
Factor out the GCF
1. 2x
^{4}  16x
^{2}
2. 4x
^{2}y
^{3} + 20xy
^{2} + 12xy
3. 2x
^{3} + 8x
^{2}  4x
4. y
^{3}  2y
^{2} + y  7
Related Topics:
Factoring Quadratic Equations using
Perfect Square Trinomial (Square of a Sum or Square of a Difference) or
Difference of Two Squares.
Factoring Quadratic Equations where the
coefficient of x^{2 }is 1.
Factoring Quadratic Equations where the
coefficient of x^{2} is greater than 1
Factoring Quadratic Equations by
Completing the Square
Factoring Quadratic Equations using the
Quadratic Formula.
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