Factoring Quadratics
2026-02-28 08:24 Diff
https://brightchamps.com/en-us/math/algebra/factoring-quadratics

Depending on the structure of the quadratic equation, it can be factored using various methods. These include: 

Factoring by splitting the middle term

The middle terms are split such that their sum equals the coefficients of x and their product equals the product of x2 and the constant term.

  • Let's begin with a quadratic equation in the form ax² + bx + c = 0.
     
  • We then find two numbers whose sum is b and whose product is equal to a × c.
     
  • The middle term is rewritten as split terms
     
  • Group the terms into pairs.
     
  • Find the GCF from each group.
     
  • Factor out the common binomial expression.

Example: Factorize 3x² + 11x + 6

  • To split the middle term, find numbers having a sum of 11 and a product of  3 × 6 = 18. 9 and 2 are the required numbers.
     
  • Rewrite the middle term: 3x² + 9x + 2x + 6
     
  • Group the terms (3x² + 9x) + (2x + 6)
     
  • Factor 3x(x + 3) + 2(x + 3)
     
  • Factor (3x + 2)(x + 3)


Therefore, 3x² + 11x + 6 = (3x + 2)(x + 3)

Factoring using the formula

We use this method when x² has the coefficient 1 and the quadratic is expressed as (x + a)(x + b).

Steps:

Write the quadratic in its standard form, i.e.,  x² + bx + c.


Find two numbers, whose sum is b and product is c.


Rewrite the quadratic using the factors.


Example: Factorize x² + 7x + 12

  • We use the numbers 3 and 4, because,  3 + 4 = 7 and 3 × 4 = 12
     
  • Factor: (x + 3)(x + 4)

Thus, x² + 7x + 12 = (x + 3)(x + 4)

Factoring using the quadratic formula


Also known as Shridharacharya’s formula, this is a universal method for solving and factorizing quadratic equations.

The formula is: x = -b  b2- 4ac2a     

Steps:

  • The equation should be written in standard form, that is, ax² + bx + c = 0
     
  • Then the quadratic formula is used for finding roots x₁ and x₂.
     
  • This gives us the factored form a(x - x₁)(x - x₂)


Example: Factorize 2x² - 5x - 3

Use the quadratic formula:
 x = - (-5)   (-5 )2 - 4(2)(-3)2(2)  =   5   25 + 244   = 5   454     = 5   74

Roots: x₁ = 3, x₂ = -½


Rewrite: 2(x - 3)(x + ½)

Thus, 2x² - 5x - 3 = 2(x - 3)(x + ½)

Now we will simplify it further.
(x - 3) (x + 12) = x2 + 12x - 3x - 32 = x2 - 52x - 32

Multiplying everything by 2, we get
2 (x2 - 52x - 32) = 2x2 - 5x - 3

So the simplified expression is 2x2 - 5x - 3

Factoring Using Algebraic Identities

This method uses well-known algebraic identities to factorize special forms of quadratic expressions quickly.

Common identities used:

  • (a + b)² = a² + 2ab + b²
     
  • (a - b)² = a² - 2ab + b²
     
  • a² - b² = (a + b)(a - b)


Steps:

  • Recognize the algebraic identity that can be applied to the given equation.
     
  • Apply the appropriate identity to factorize.


Example: Factorize 9x² - 16

This equation is a difference of squares: (3x)² - (4)²

So, we apply the identity: a² - b² = (a + b)(a - b),

This gives us, 9x² - 16 = (3x + 4)(3x - 4)