Equations and Inequalities

This chapter is about solving systems and ranges of values — and presenting the answer in the exact form the mark scheme wants.

Simultaneous equations

When one equation is linear and one is quadratic, use substitution: rearrange the linear equation for one variable and put it into the quadratic.

Worked example. Solve $y = x$ and $y = x^2 - 2$. Substituting gives $x = x^2 - 2$, so $x^2 - x - 2 = 0$, i.e. $(x-2)(x+1) = 0$. Then $x = 2, y = 2$ or $x = -1, y = -1$.

Graphically, the solutions are the points where the line meets the curve.

Exam link. If the line is a tangent to the curve, the resulting quadratic has a repeated root — so its discriminant is zero. This connects straight back to the Quadratics chapter.