Apply quadratic theory: solve quadratics by formula and completing the square, use the discriminant to determine the nature of the roots, and solve quadratic inequalities.

What this dot point is asking

Quadratic theory in CCEA GCSE Further Mathematics goes beyond solving equations to understanding them. You must complete the square, use the discriminant to decide how many real roots an equation has (and find unknown constants from that condition), solve simultaneous equations where one is quadratic, and solve quadratic inequalities. These ideas connect the algebra to graphs and to calculus, and the discriminant in particular is a recurring source of exam marks.

Completing the square

Completing the square rewrites a quadratic so the variable appears once, which both solves the equation and exposes the turning point of the parabola. For $x^2 + bx + c$, halve the $x$ coefficient, square the bracket, then subtract the square you added.

For example $x^2 + 8x + 3 = (x + 4)^2 - 16 + 3 = (x + 4)^2 - 13$. The turning point of $y = x^2 + 8x + 3$ is therefore $(-4, -13)$. When the leading coefficient is not $1$, factor it out of the $x^2$ and $x$ terms first, complete the square inside, then multiply back.

The discriminant and the nature of the roots

The discriminant is the part of the quadratic formula under the root sign, and its sign alone tells you how many real solutions exist without solving.

Many questions give a condition, such as "equal roots" or "no real solutions", and ask for an unknown constant. Translate the condition into an equation or inequality in the discriminant, then solve it. This is one of the most common Further Mathematics question types.

Simultaneous equations with a quadratic

When one equation is linear and one is quadratic, substitute the linear equation into the quadratic and solve the resulting single quadratic.

Quadratic inequalities

To solve a quadratic inequality, first solve the matching equation to find the critical values, then think about the parabola's shape. A positive parabola is below the axis between its roots and above the axis outside them.

For $x^2 - x - 6 > 0$, factorise to $(x - 3)(x + 2) > 0$, giving critical values $x = 3$ and $x = -2$. The parabola opens upward, so it is positive outside the roots: the solution is $x < -2$ or $x > 3$. For the reverse inequality $\le 0$, the solution would be the region between the roots, $-2 \le x \le 3$. Sketching the parabola is the safest way to read off the correct region.

Why this matters

Quadratic theory links algebra to graphs and calculus. Completing the square gives the vertex you also find by differentiation; the discriminant decides whether a line cuts, touches, or misses a curve, which matters for tangents and for the circle topic; and quadratic inequalities appear when you analyse where a function is positive. Mastery here underpins functions, coordinate geometry and the applied units too.