Learn: Factorising Quadratic Expressions

Welcome!Today we'll learn about factorising quadratic expressions. This topic is essential in algebra and helps us simplify and solve equations effectively!

What is a quadratic expression?A quadratic expression is a polynomial expression where the highest power of the variable is 2. For example, x² + 5x + 6 is a quadratic expression.

What does factorising mean?Factorising means breaking down an expression into simpler 'factors' that can be multiplied together to give the original expression. For quadratics, this usually involves finding two expressions in brackets, such as (x + a)(x + b).

Why is factorisation useful?Factorisation allows us to solve quadratic equations by finding the values of x that make the equation equal zero. It's also useful for simplifying expressions in algebra.

How to factorise a simple quadratic?To factorise a quadratic, you look for two numbers that multiply to give the constant term and add to give the coefficient of the middle term. For example, in x² + 5x + 6, the two numbers are 2 and 3 because 2 × 3 = 6 and 2 + 3 = 5.

Steps to factorise1. Write the quadratic in standard form: ax² + bx + c.2. Find two numbers that multiply to give the constant term (c) and add to give the coefficient of x (b).3. Split the middle term into two terms using the numbers found in step 2.4. Factorise each pair of terms.5. Write the final factorised form as two brackets.

Factorising harder quadraticsSometimes you may need to factorise quadratics where the coefficient of x² is not 1. For example, 2x² + 5x + 3. You can use the same method, but it involves an extra step.

Example: Factorise 2x² + 5x + 31. Multiply the coefficient of x² (2) by the constant term (3) to get 6.2. Find two numbers that multiply to give 6 and add to give 5. These numbers are 2 and 3.3. Split the middle term: 2x² + 2x + 3x + 3.4. Factorise each pair: 2x(x+1) + 3(x+1).5. Combine: (2x+3)(x+1).

Special cases in factorising quadraticsSome quadratics involve difference of squares, which are written as (a² - b²). These can be factorised into (a+b)(a-b). For example, x² - 9 becomes (x+3)(x-3).

Review Time!Great work! You've learned about quadratic expressions, basic factorisation, and special cases like difference of squares. Now let's test your understanding.

Well Done!You’ve completed this lesson on factorising quadratic expressions. Remember to practise these steps regularly to master them!