Factoring Quadratics

Factorization of a quadratic equation is a key algebraic technique where a quadratic expression is rewritten as the product of two binomials. This method simplifies solving quadratic equations, rewriting algebraic expressions, and understanding the geometric properties of quadratic functions, such as their x-intercepts.

Methods to Factorize Quadratic Equations

There are several methods to factorize quadratic equations, each suited for different types of equations. The primary methods are:

Factoring Using Splitting Middle-Term

This method splits the middle term into two parts that add up to the middle coefficient and multiply to the product of the first and last coefficients.

• Step 1: Identify the quadratic equation ax² + bx + c = 0.
• Step 2: Find two numbers whose sum is b (the coefficient of x) and whose product is ac (the product of the coefficients of x² and the constant term).
• Step 3: Rewrite the middle term using these two numbers.
• Step 4: Group the terms into two pairs.
• Step 5: Factor out the common factor from each pair.
• Step 6: Factor out the common binomial factor from the resulting expression.

Example: Factorize 2x² + 7x + 3.

Identify the quadratic equation: 2x² + 7x + 3

Find two numbers: We need two numbers that add to 7 and multiply by 2 * 3 = 6. These numbers are 6 and 1

Rewrite the middle term: 2x² + 6x + x + 3

Group the terms: (2x² + 6x) + (x + 3)

Factor out the common factor: 2x(x + 3) + 1(x + 3)
Factor out the common binomial factor: (2x + 1)(x + 3)

Thus, 2x² + 7x + 3 = (2x + 1)(x + 3)

Factoring Quadratic Equations using Formula

This method uses a specific formula to factorize the quadratic equation.

• Step 1: Identify the quadratic equation in the form ax² + bx + c.
• Step 2: Use the factorization formula: If a ≠ 1, use the formula for ax² + bx + c.
• Step 3: Rewrite the equation based on the identified formula.
• Step 4: Simplify to find the factors.

Example: Factorize x² + 5x + 6.

Identify the quadratic equation: x² + 5x + 6
Use the factorization formula: x² + (a + b)x + ab = (x + a)(x + b)

Here, a = 2 and b = 3, since 2 + 3 = 5 and 2 × 3 = 6
Rewrite the equation: (x + 2)(x + 3)

Thus, x² + 5x + 6 = (x + 2)(x + 3).

Factoring Quadratic Equation using Quadratic Formula

Quadratic formula also known as Shreedhara Acharya’s Formula is a standard method for solving any quadratic equation and can also be used for factorization. The quadratic formula is derived from the process of completing the square and is given by:

This formula gives the roots of the quadratic equation, which can then be used to factorize the quadratic expression.

• Step 1: Identify the quadratic equation in the form ax² + bx + c
• Step 2: Apply the quadratic formula to find the roots x₁ and x₂
• Step 3: Rewrite the quadratic equation as a(x - x₁)(x - x₂)

Example: Factorize 3x² - 2x - 8.

Identify the quadratic equation: 3x² - 2x - 8
Apply the quadratic formula: x = (-(-2) ± √((-2)² - 4.3.(-8)) / (2.3)

Simplify the discriminant
x = x = (2 ± √(4 + 96)) / 6 = x = (2 ± √(100)) / 6 = x = (2 ± 10) / 6

Find the roots: x₁ = 2 and x₂ = - 4/3
Rewrite the equation: 3 (x - 2) (x + 4/3)

Thus, 3x² - 2x - 8 = 3 (x - 2) (x + 4/3)

Factoring Quadratic Equations using Algebraic Identities

Algebraic identities can simplify the factorization process, particularly when dealing with specific forms of quadratic equations. They provide a straightforward method to factorize certain types of quadratic expressions.

• Step 1: Identify the quadratic equation in a recognizable identity form.
• Step 2: Apply the appropriate identity to factorize the expression.

Example: Factorize 4x² - 9.

Identify the quadratic equation: 4x² - 9 is in the form a² - b²
Apply the identity: a² - b² = (a + b)(a - b), where a = 2x and b = 3
Rewrite the equation: (2x + 3)(2x - 3)

Thus, 4x² - 9 = (2x + 3)(2x - 3)

Related Articles:

• Quadratic Function
• Quadratic Inequalities
• Solving Cubic Equations

Solved Examples of Factorization of Quadratic Equations

Example 1: Factorize: x² - 5x + 6.

To factorize: x² - 5x + 6

⇒ x² - 2x - 3x + 6
⇒ x(x - 2) - 3(x - 2)
⇒ (x - 2)(x - 3)

Example 2: Factorize: x² - 6x + 9.

x² - 6x + 9
⇒ x² - 2.3x + (3)²

Comparing with a² - 2ab + b² = (a - b)²
⇒ (x - 3)²

Example 3: Factorize: x² + x - 12.

x² + x - 12

⇒ x² +4x - 3x + 12
⇒ x(x + 4) - 3(x + 4)
⇒ (x + 4)(x - 3)

Example 4: Factorize: x² + 8x + 16.

x² + 8x + 16
⇒ x² + 2.4x + (4)²

Comparing with a² + 2ab + b² = (a + b)²
⇒ (x + 4)²