Solving quadratic equations by factoring is one of the methods to find a solution for the roots of the given polynomial equation. It is also one of the easiest ways to solve quadratic equations. A polynomial equation is an algebraic equation with different terms like variables with varying coefficients, powers, and constants. There happens to be only one variable in a polynomial equation, but the variable has different coefficients within the equation. The equation contains different operations like addition, subtraction, multiplication, and division.

Quadratic Equation

  • A quadratic polynomial equation has the highest power on the variable as 2. 

  • Thus, a quadratic polynomial equation results in terms of the square root of the polynomial. 

  • The equation has one variable (say, ‘x’) accompanied by various coefficients and powers (say, 3x²+ 13x). 

  • A quadratic equation is always in the format ax² + bx + c.

  • The term ‘a’ in a quadratic equation is always ‘non-zero’.

Solving Quadratic Equation by Factoring

A quadratic polynomial equation can be solved by one of the three basic methods. 

  • Factoring

  • Using the quadratic formula 

  • Completing the square

Here, we will be discussing the factoring method. The factors of a number are a pair of smaller numbers that give the larger value of multiplying with each other. For example:

The factors of 15 are 5 and 3.

The factors of 16 are 8, 2 and 4.

Procedure of Solving

Any given quadratic equation can be solved by the factoring method. Consider an equation 3x² + 13x + 4. It is a quadratic polynomial equation.

  1. Split the middle term

Split the middle term (13x) into two terms in such a way that 

  • The sum of the two numbers gives the middle term value (13) 

  • The product of the two numbers is equal to the product of the coefficient of x² is 3 and the constant 4, that is 12×2.

The middle term(13x) is split up into 2 terms (12x and 1x)

3x² + 13x + 4 

3x² + 12x + 1x + 4

  1. Take common terms out

Group two terms that have common terms. This will result in the terms inside the parentheses being the same.

3x²+12x+1x+4

  • In the first and second terms (3x² and 12x, respectively), 3x is common.

  • In the third and fourth terms (1x and 4 respectively), 1 is the common term.

  • Take the common terms out

3x (x+4) + 1 (x+4)

  • Again, from the resulting equation, take a common term out, grouping the previously taken common terms.

3x (x+4) + 1 (x+4)

= (x+4) × (3x+1)

  • The common term(x+4) has been taken out from the equation.

  • So, the earlier taken common terms (3x and 1) are grouped.

  1. Equate the terms to find the root

  • Equate the terms within both the parentheses, separately, to zero.

Taking x+4 = 0

x = -4

Taking 3x+1 = 0,

3x = -1

x = -1/3

x values are the factors

x = -4 or -1/3

Solved Examples of Quadratic Equations by Factoring Method

A few solved examples of quadratic equations by the factoring method are given below:

Solve the quadratic equation 3x² – 2x – 8 by the factoring method.

3x² – 2x – 8

= 3x² + 4x – 6x – 8

= x (3x + 4) – 2 (3x + 4)

= (3x + 4) (x – 2)

Equating 3x + 4,

3x + 4 = 0

3x = -4

x = -4/3

Equating x – 2,

x – 2 = 0

x = 2

Answer: x = -4/3 or 2.

Solve the quadratic equation 6x² + 11x – 35 by the factoring method.

6x² + 11x – 35

= 6x² + 21x – 10x – 35

= 3x (2x + 7) – 5 (2x + 7)

= (2x + 7) (3x – 5)

Equating 2x + 7,

2x + 7 = 0

2x = -7

x = -7/2

Equating 3x – 5,

3x – 5 = 0

3x = 5

x = 5/3

Answer: x = -7/2 or 5/3.

Solve the quadratic equation 2x² – 3x – 5 by the factoring method.

2x² – 3x – 5

= 2x² + 2x – 5x – 5

= 2x (x + 1) – 5 (x + 1)

= (x + 1) (2x – 5)

Equating x + 1,

x + 1 = 0

x = -1

Equating 2x – 5 = 0,

2x – 5 = 0

2x = 5

x = 5/2

Answer: x = -1, 5/2.

Solve the quadratic equation 2(x² + 1) = 5x by the factoring method.

2(x² + 1) = 5x

Simplifying and rearranging the equation to move the terms to left:

= 2x² + 2 – 5x

= 2x² – 5x + 2

= 2x² – 4x – x + 2

= 2x (x – 2) – 1 (x – 2)

= (x – 2) (2x – 1)

Equating x – 2 = 0,

x = 2

Equating 2x – 1 = 0,

2x = 1

x = 1/2

Answer: x = 2, 1/2.

Solve the quadratic equation x² – 3x – 4 by the factoring method.

x² – 3x – 4

= x² + x – 4x – 4

= x (x + 1) – 4 (x + 1)

= (x + 1) (x – 4)

Equating x + 1,

x = -1

Equating x – 4,

x = 4

Answer: x = -1, 4.

Conclusion

A quadratic equation has the highest exponential as 2. Thus, they are named ‘quadratic’. Solving quadratic equations by the factoring method is only one of the methods to solve a quadratic polynomial equation. There are two more methods to solve a quadratic equation. Those methods are the quadratic formula method and completing the square method.