Factorising Quadratics - GCSE Maths

Practice Questions
Downloads

Introduction

  • Quadratic equations are one of the most important topics in algebra.
  • They help us solve problems involving quantities that change in a curved or parabolic pattern.

Quadratic Equations are used in:

  • Real world Application:

Examples of solving quadratic equations in real life: projectile motion of a basketball and computer graphics rendering

  • In Trigonometry, Probability, and Statistics.
  • Used in polynomial functions and complex numbers.
  • Essential for calculus (finding maxima/minima, derivatives).
  • Helps in understanding graphs of functions (parabolas, vertex, axis of symmetry).

What are Quadratic Equations?

  • A quadratic equation is a second-degree polynomial equation in a single variable x, typically written in the standard form:

Standard form of a quadratic equation ax squared plus bx plus c equals zero

where,

    • a, b, and c are constants with a ≠ 0.
    • x represents the unknown variable.

Key Features:

Degree:

  • The highest power of x is 2, making it a Quadratic Equation.

Graphical Representation:

  • The graph of a quadratic equation is a parabola, if a > 0 then parabola opens upward and if a < 0 then it opens downward.

Graphs showing parabolas for when a is greater than zero and a is less than zero in quadratic equations

How to solve Quadratic Equations using Factorization?

  • Factoring expresses the equation as a product of two terms (if possible).
  • Methods to Solve Factorising Quadratics:

When the x² Coefficient is 1:

  • Step#1: Identify coefficients.
  • Step#2: Find two numbers that multiply to a × c and add to b.
  • Step#3: Factor out the common term.
  • Step#4: Solve for x.

When the Coefficient of x² is More Than 1:

  • Step#1: Identify coefficients:
  • Step#2: Multiply the coefficient of x² and the constant term.
  • Step#3: Find two numbers that multiply to a × c and add to b.
  • Step#4: Rewrite the middle term using these numbers.
  • Step#5: Factor by grouping.
  • Step#6: Factor out the common term.
  • Step#7: Solve for x.

certified Physics and Maths tutorSolved Example

Problem: Solve the quadratic equation by factoring:

x2 − 5x + 6 = 0

Solution: 

Step #1: Identify coefficients:

Here:

    • a = 1 (coefficient of x2)
    • b = −5 (coefficient of x)
    • c = 6 (constant term)

Step #2: Find two numbers that multiply to a × c and add to b:

    • The numbers are -2 and -3

Because:

Quadratic equation x squared minus 2x minus 3x plus 6 equals 0

Step #3: Factor out the common term:

Factorising Quadratics equation using grouping and common factors

Step #4: Solve for x:

Solving x - 2 = 0 and x - 3 = 0 to find quadratic equation roots

The Solution are x = 2 and x = 3

Final Answer: x = 2 and x = 3

certified Physics and Maths tutorSolved Example

Problem: Factor the quadratic expression:

2x2 + 7x + 3

Solution: 

Step #1: Identify coefficients:

Here:

    • a = 2 (coefficient of x2)
    • b = 7 (coefficient of x)
    • c = 3 (constant term)

Step #2: Multiply the coefficient of x2 and the constant term.

2 times 3 equals 6 shown as part of solving quadratic equation

Step #3: Rewrite the middle term using these numbers:

    • The numbers are 6 and 1

Because:

6 times 1 equals 6 and 6 plus 1 equals 7 showing factor pair for quadratic equation

Step #4: Rewrite the middle term using these numbers:

Quadratic expression 2x squared plus 6x plus x plus 3 before factorisation

Step #5: Factor by grouping:

Grouping and Factorising Quadratics 2x squared plus 6x and x plus 3 in a quadratic equation

Step #6: Factor out the common term:

Factored quadratic equation shown as (2x + 1)(x + 3) equals zero

Step #7: Solve for x:

Solving quadratic equation by setting each factor to zero and finding x equals -1/2 or -3

The Solution are x = -1/2 and x = -3

Final Answer: x = -1/2 and x = -3

certified Physics and Maths tutorSolved Example

Problem: Factor the quadratic expression:

2x2 + 9x + 7

Solution: 

Step #1: Identify coefficients:

Here:

    • a = 2 (coefficient of x2)
    • b = 9 (coefficient of x)
    • c = 7 (constant term)

Step #2: Multiply the coefficient of x2 and the constant term.

Example showing 2 times 7 equals 14 as part of quadratic equation solving

Step #3: Rewrite the middle term using these numbers:

    • The numbers are 2 and 7

Because:

2 times 7 equals 14 and 2 plus 7 equals 9 example used in solving quadratic equations

Step #4: Rewrite the middle term using these numbers:

Expression showing 2x squared plus 2x plus 7x plus 7 for quadratic equation

Step #5: Factor by grouping:

Quadratic equation showing grouped terms 2x squared plus 2x and 7x plus 7

Step #6: Factor out the common term:

Factorising Quadratics equation in the form (2x + 7)(x + 1) equals 0

Step #7: Solve for x:

Solving the quadratic equation shows x equals negative seven over two or x equals negative one

The Solution are x = -7/2 and x = -1

Final Answer: x = -7/2 and x = -1

certified Physics and Maths tutorSolved Example

Problem: Solve the quadratic equation by factoring:

x2 − 8x + 15 = 0

Solution: 

Step #1: Identify coefficients:

Here:

    • a = 1 (coefficient of x2)
    • b = 8 (coefficient of x)
    • c = 15 (constant term)

Step #2: Find two numbers that multiply to a × c and add to b:

    • The numbers are 3 and 5

Because:

Using the pair 3 and 5 to factor a quadratic equation with x squared plus 8x plus 15

Step #3: Factor out the common term:

Factorising Quadratics x squared plus 8x plus 15 into two brackets using common terms

Step #4: Solve for x:

Solving quadratic equation by equating each factor to zero: x = -3 or x = -5

The Solution are x = -3 and x = -5

Final Answer: x = -3 and x = -5

Request for Demo Lesson

Table of Content

Worksheet on Factorising Quadratics

Question 1: Factorize x2 + 10x + 25

)

Question 2: Factorize x2 + 9x + 14

)

Question 3: Factorize 2x2 + 17x + 36

)

Question 4: Factorize 5x2 + 62x + 24

)

Question 5: Factorize 7x2 + 10x + 3

)