Quadratic inequalities are precisely what they sound like: inequalities that involve a squared term, and therefore a quadratic. Much like normal quadratics, they usually have two-part solutions rather than just one. As with linear equalities, we can manipulate them to find solutions as if they were equations, but in the case of a quadratic inequality this gets a little more complicated. Also, depending on which sign is used in your inequality, your solutions can look quite different.
Before going further, you should be familiar both with linear inequalities (https://mathsmadeeasy.co.uk/gcse-maths-revision/inequalities-number-line-solving-inequalities/) and with using factorisation to solve quadratics (https://mathsmadeeasy.co.uk/gcse-maths-revision/solving-quadratics-factoring/). Now, let’s get into it.
Example: Solve the inequality x^2-2x-3<0.
The first step is to factorise this quadratic and find the solutions as if it were an equation, not an inequality. So, observing that 1\times(-3)=-3 and 1+(-3)=-2, we get that x^2-3x-3=(x+1)(x-3) and so our inequality becomes
So, if this were an equation, then the solutions to the quadratic would be x=-1 and x=3. Using this information, we can sketch the graph of y= x^2-2x-3 – this is the step that will help us understand how to solve the inequality.