Quadratic Equations (H)

Quadratic expressions are recognisable as they have an x² term along with (usually) an x-term and a number. Quadratic equations don’t behave like linear ones – sometimes they don’t even have a solution, yet at other times they can have 2! Quadratic curves, called parabolas, occur in nature and in real-life situations, so it’s a good idea to know all the intricacies of them. See how well you do in this GCSE Maths quiz!

Quadratics are the first in a series of functions called POLYNOMIALS. They all have decreasing powers of x, where the power is always a whole number. The graphs of polynomials are all curves. The curve of a quadratic can be described as U-shaped (or ∩-shaped for negative x²), which means its gradient is always changing. Compare this to the straight line graph, which has a fixed gradient.

You should be able to carry out a few different techniques with quadratic expressions. Factorising a quadratic is the first step to solving quadratic equations, but remember not all quadratics will factorise! The ‘perfect square’ and the ‘difference of two squares’ are two special cases, and you should learn how to recognise them.

The roots of a quadratic curve, which is where it crosses the x-axis, correspond to the solutions of the quadratic equation.

Completing the square will allow you to identify the coordinates of the turning point. You can complete the square on any quadratic expression, even if it doesn’t cross the x-axis.

For quadratics that look a bit tricky, use the quadratic formula. The discriminant will give you an indication of the nature of any roots, which will tell you if a solution exists or not.

Question 1