How to Solve by Factorisation

Solving quadratic equations is a fundamental skill in algebra that often appears in various mathematical problems and real-world applications. One of the most effective and widely used methods to solve quadratic equations is by factorisation. This technique involves rewriting the quadratic expression as a product of simpler binomials, making it easier to find the solutions. Mastering factorisation not only simplifies solving equations but also deepens your understanding of polynomial expressions and their properties. In this guide, we will explore how to solve quadratic equations through factorisation, step by step, with practical tips and examples to help you become confident in applying this method.

How to Solve by Factorisation

Understanding Quadratic Equations

Before diving into the process of factorisation, it’s essential to understand what a quadratic equation is. A quadratic equation is a second-degree polynomial equation of the form:

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0. The solutions to this equation are the values of x that make the expression equal to zero. These solutions are also called roots or zeros of the quadratic function.

There are various methods to solve quadratic equations, including completing the square and the quadratic formula. However, factorisation is often the quickest way when the quadratic can be factored easily.

Steps to Solve Quadratic Equations by Factorisation

Follow these systematic steps to solve a quadratic equation via factorisation:

1. Write the quadratic equation in standard form

Ensure the quadratic is written as ax² + bx + c = 0. If it’s not, rearrange the terms accordingly.

Example: 2x² + 7x - 3 = 0

2. Find two numbers that multiply to a * c and add to b

Identify two numbers that satisfy:

• Product = a × c
• Sum = b

This step is crucial because these two numbers will help split the middle term and facilitate factorisation.

Example: For 2x² + 7x - 3:

• a × c = 2 × (-3) = -6
• b = 7

Find two numbers that multiply to -6 and add to 7: these are 6 and -1 because:

• 6 × -1 = -6
• 6 + (-1) = 5 (which is not 7, so check again)

Actually, for 7, the numbers are 6 and 1, but their product is 6, not -6. So, look carefully:

• Numbers: -1 and 6
• -1 × 6 = -6
• -1 + 6 = 5 (not 7)

Wait, perhaps the quadratic is different. Let's pick a different example for clarity:

Example: Solve x² + 5x + 6 = 0

• a = 1, b = 5, c = 6
• a × c = 6
• Numbers that multiply to 6 and add to 5: 2 and 3

3. Rewrite the middle term using these two numbers

Split the middle term into two terms using the numbers found:

For x² + 5x + 6:

Rewrite as: x² + 2x + 3x + 6

4. Factor by grouping

Group the terms into two pairs and factor each group:

(x² + 2x) + (3x + 6)

Factor out common factors from each group:

x(x + 2) + 3(x + 2)

Notice (x + 2) is common, so factor it out:

(x + 2)(x + 3) = 0

5. Solve for x

Set each binomial equal to zero and solve:

• x + 2 = 0 ⇒ x = -2
• x + 3 = 0 ⇒ x = -3

These are the solutions to the quadratic equation.

Special Cases and Tips for Factorisation

While factorisation is straightforward for some quadratics, certain cases require extra attention:

• Difference of squares: Recognize patterns like a² - b² = (a - b)(a + b)
• Perfect square trinomials: e.g., a² + 2ab + b² = (a + b)²
• Quadratics that do not factor easily: If you cannot find suitable numbers or the quadratic is prime, consider using the quadratic formula instead.

Example of difference of squares:

Solve x² - 16 = 0

Factor as:

(x - 4)(x + 4) = 0

Solutions:

• x - 4 = 0 ⇒ x = 4
• x + 4 = 0 ⇒ x = -4

Tips for successful factorisation:

• Always write the quadratic in standard form first.
• Look for common factors in all terms before attempting to factor further.
• Practice identifying patterns like difference of squares or perfect square trinomials.
• If the quadratic cannot be factored easily, use the quadratic formula as a backup method.

Practice Examples

Let’s look at a few practice problems to reinforce the process:

Example 1:

Solve x² + 3x - 4 = 0

• a = 1, b = 3, c = -4
• a × c = -4
• Numbers that multiply to -4 and add to 3: 4 and -1

Rewrite as:

x² + 4x - x - 4

Factor by grouping:

x(x + 4) - 1(x + 4)

(x - 1)(x + 4) = 0

Solutions:

• x = 1
• x = -4

Example 2:

Solve 3x² + 12x + 12 = 0

• Divide the entire equation by 3 to simplify:

x² + 4x + 4 = 0

This is a perfect square trinomial:

(x + 2)² = 0

Solution:

x + 2 = 0 ⇒ x = -2

Summary of Key Points

Solving quadratic equations by factorisation is a powerful technique that relies on rewriting the quadratic as a product of binomials. The main steps involve writing the quadratic in standard form, finding two numbers that multiply to a_c and add to b, splitting the middle term, and factoring by grouping. Recognizing patterns such as difference of squares and perfect square trinomials can make the process more efficient. Practice with different types of quadratics will improve your ability to solve them quickly and accurately. When factorisation is not straightforward, remember that the quadratic formula is a reliable alternative. Mastering these methods enhances your algebraic skills and prepares you for more advanced mathematical concepts.