Quadratic Functions and Equations

The Quadratic Formula

The solutions (roots) of $ax^2 + bx + c = 0$ are given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula is derived by completing the square on the general quadratic and is guaranteed to find the roots whenever they exist. You should memorise this formula — it appears on the AQA and OCR formula sheets but not always on Edexcel's.

The Discriminant

The expression $\Delta = b^2 - 4ac$ is called the discriminant. It determines the nature of the roots:

• $\Delta > 0$: two distinct real roots
• $\Delta = 0$: one repeated real root (the parabola touches the $x$-axis)
• $\Delta < 0$: no real roots (the parabola does not cross the $x$-axis)

The discriminant is a powerful tool for answering questions about whether a line intersects a curve, whether an equation has solutions, or whether a quadratic is always positive (or always negative).

Completing the Square

Any quadratic $ax^2 + bx + c$ can be written in the form $a(x + p)^2 + q$. This is called completing the square and reveals the vertex (turning point) of the parabola.

For $f(x) = x^2 + bx + c$:

$f(x) = \left(x + \frac{b}{2}\right)^2 - \frac{b^2}{4} + c$

For $f(x) = ax^2 + bx + c$ where $a \neq 1$, first factor out $a$ from the first two terms:

$f(x) = a\left(x^2 + \frac{b}{a}x\right) + c = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c$

The vertex is at $\left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right)$.

Completing the square also allows you to express the minimum (or maximum) value of a quadratic function, which is essential for optimisation problems.

Factorising Quadratics

A quadratic $ax^2 + bx + c$ can be factorised if and only if $\Delta \geq 0$. Methods include:

• Inspection: Find two numbers that multiply to give $ac$ and add to give $b$.
• Grouping: For $a \neq 1$, split the middle term using the two numbers found above.
• Using roots: If the roots are $\alpha$ and $\beta$, then $ax^2 + bx + c = a(x - \alpha)(x - \beta)$.

Sketching Parabolas

To sketch $y = ax^2 + bx + c$, identify:

1. Shape: $a > 0$ gives a U-shape; $a < 0$ gives an inverted U.
2. $y$-intercept: Set $x = 0$ to get $y = c$.
3. $x$-intercepts (roots): Solve $ax^2 + bx + c = 0$ using factorisation, completing the square, or the quadratic formula.
4. Vertex (turning point): Use completed square form or $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$.
5. Line of symmetry: $x = -\frac{b}{2a}$.

Quadratic Inequalities

To solve an inequality such as $ax^2 + bx + c > 0$:

1. Find the roots of $ax^2 + bx + c = 0$.
2. Sketch the parabola (you only need a rough sketch).
3. Read off the solution from the sketch.

For $a > 0$:

• $ax^2 + bx + c > 0$ is satisfied where the parabola is above the $x$-axis (outside the roots): $x < \alpha$ or $x > \beta$.
• $ax^2 + bx + c < 0$ is satisfied where the parabola is below the $x$-axis (between the roots): $\alpha < x < \beta$.

For $a < 0$, the regions are reversed.

The Intersection of a Line and a Curve

To find where a line $y = mx + d$ meets a curve $y = ax^2 + bx + c$, set them equal:

$ax^2 + bx + c = mx + d$

$ax^2 + (b - m)x + (c - d) = 0$

The discriminant of this equation tells you about the intersection:

• $\Delta > 0$: the line cuts the curve at two points
• $\Delta = 0$: the line is tangent to the curve
• $\Delta < 0$: the line does not meet the curve

Tip 1: Know When to Use Each Method

Factorisation is quickest when it works. The quadratic formula always works for real roots. Completing the square is best when you need the vertex or must express a minimum/maximum value.

Tip 2: Sketch Before Solving Inequalities

A quick sketch of the parabola makes inequalities straightforward. Without it, students often write the wrong regions or mix up strict and non-strict inequalities.

Tip 3: Use the Discriminant as a Condition

Many A-Level questions ask you to find values of a parameter $k$ such that an equation has certain properties (e.g., "no real roots" or "a repeated root"). Set up the discriminant condition and solve the resulting inequality or equation.

Tip 4: Present Completed Square Form Neatly

When completing the square, show each step clearly. Examiners award method marks for the intermediate stages, so even if you make an arithmetic error, you can still earn most of the marks.

Tip 5: Check with Substitution

After solving a quadratic equation, substitute your roots back into the original equation to verify they satisfy it. This takes only a few seconds and catches sign errors.