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# Solving Quadratic Equations by Factoring

A quadratic equation contains a variable term with the exponent of 2 and no variable term with a higher power. A quadratic equation with one variable, x is an equation contains x2 as the highest power of x. A quadratic equation can have at most two solutions. A quadratic equation is also called a second-degree equation.

A quadratic equation is of the form ax2 + bx + c = 0, a $\neq$ 0. where a, b and c are real numbers.

The general methods used to solve quadratic equations are:
1. Factoring method
2. Taking square root method
3. Completing the square
4. Using Quadratic formula
5. Graphical method

## Steps Involved in Factoring Quadratic Equations

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The algebraic expression in the equation can be factored any of the methods applied for factoring quadratic expressions like, GCF method, splitting the middle term method or using binomial identities. The two solutions of the quadratic equation are obtained by equating the factors to zero using zero factor property and solving the linear equations so formed. The steps involved in factoring a quadratic equation of the form ax2 + bx  + c = 0 are,

Steps for Solving Quadratic Equations:

Step 1: Determine the values of a, b and c.

Step 2: Find the product "a * c".

Step 3: Choose any two numbers b1 and b2 such that b1b2 = a * c and b1 + b2 = b.

Step 4: Rewrite the equation as ax2 + (b1)x + (b2)x + c = 0

Step 5: Factor out the GCF pairwise.

Step 6:
Repeat factoring using GCF.

## Factoring a Quadratic Equation Using GCF

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Before solving quadratic equations by using factorized method, let's see some examples for solving quadratic equations by GCF.

### Solved Examples

Question 1: Solve the quadratic equation 2x2- 4x = 0
Solution:
The Greatest common factor of the two terms 2x2 and 4x  is 2x.

Reversing the distribution in the expression 2x2 - 4x by pulling 2x out , the equation is written in factored form as,

2x(x - 2) = 0

Applying zero factor property,

either 2x = 0 or x - 2 = 0

=> x = 0 or x = 2

=> Solutions of the quadratic equation as x = 0 and x = 2

Question 2: Solve the quadratic equation 3x2- 9x = 3x
Solution:
Given, quadratic equation 3x2- 9x = 3x

=> 3x2 - 9x - 3x = 0

=> 3x2 -12x = 0

The Greatest common factor of the two terms 3x2 and 12x  is 3x

=> 3x2 -12x = 3x(x - 4)

=> either 3x = 0 or x - 4 = 0

=> x = 0 or  x = 4.

## Factoring a Quadratic Equations

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The zero factor property of a product states that if the product of two or more factors is zero, then at least one of the factors is equal to zero. This property is used in solving quadratic equations by factoring. A quadratic equation is a quadratic expression with an equal sign attached. The most commonly used technique for solving these equations is factoring. This method is also known an splitting the middle term method.

The principle of zero product can be used in solving quadratic equations. Zero product states that if the product of two factors is zero, then at least one of the variable must be zero.

If a . b = 0, then a = 0 or b = 0.

### Solved Examples

Question 1: Solve the quadratic equation 2x2- 7x + 6 = 0
Solution:
Given quadratic equation 2x2- 7x + 6 = 0

Step 1:
a = 2 , b = -7 and c = 6

Step 2: ac = 2 x 6 =12

Step 3: The numbers for split are -4 and -3 as their sum (-4) + (-3) = -7 = b   and their product (-4)(-3) = 12 = ac.

Step 4: The equation is rewritten as  2x2- 4x - 3x + 6 = 0

Step 5: The terms are paired and factored using GCF method

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

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

Step 6: The common factor x - 2 is taken out and the distribution reversed

Applying zero factor property to the factored equation,

either x - 2 = 0  or 2x - 3 = 0

=> x = 2 or x = $\frac{3}{2}$

Solving the binomials, the two solutions of the given equation are x = 2 and x = 1.5

Question 2: Solve the quadratic equation x2- 5x + 6 = 0
Solution:
Given, quadratic equation x2- 5x + 6 = 0

Factor the left hand side of the equation,

=> x2- 5x + 6 = x2- 2x - 3x + 6

= x(x - 2) - 3(x - 2)

= (x - 3)(x - 2)

=> x2- 5x + 6 = (x - 3)(x - 2)

Applying zero factor property to the factored equation,

either, x - 3 = 0  or  x - 2 = 0

=> x = 3 or x = 2

=> Solutions of the given quadratic equation are x = 2, 3.