Quadratic Equations

How do you solve quadratics by factorisation?

Factorisation is the fastest method when the quadratic splits cleanly into two brackets. It works on both Standard and Higher tiers and is usually worth 3–4 marks. Follow these three steps:

1. Step 1 — Rearrange to = 0: Make sure the equation is in the form ax² + bx + c = 0. Move all terms to one side.
2. Step 2 — Factorise into two brackets: Find two numbers that multiply to give ac and add to give b. Use these to split the middle term and group, or write the brackets directly for simple quadratics (a = 1).
3. Step 3 — Set each bracket equal to zero: If (x + p)(x + q) = 0, then x + p = 0 or x + q = 0, giving x = −p or x = −q.

This method relies on the zero product property: if two things multiply to give zero, at least one of them must be zero.

What is the quadratic formula and the discriminant?

When a quadratic does not factorise neatly, the quadratic formula always works. For ax² + bx + c = 0:

x = (−b ± √(b² − 4ac)) / 2a

The expression under the square root — b² − 4ac — is called the discriminant. It tells you how many real solutions the equation has:

• b² − 4ac > 0: Two distinct real solutions (the parabola crosses the x-axis twice)
• b² − 4ac = 0: One repeated real solution (the parabola just touches the x-axis)
• b² − 4ac < 0: No real solutions (the parabola does not cross the x-axis)

This is a Higher tier topic. You must substitute a, b and c carefully — pay special attention to negative values, particularly when b is negative.

How does completing the square work?

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This is useful for two things: solving quadratics and finding the turning point of a parabola (the minimum or maximum is at (−p, q)).

For x² + bx + c (when a = 1), the process is:

1. Step 1: Halve the coefficient of x. If the coefficient is b, you get b/2.
2. Step 2: Write (x + b/2)² and then subtract (b/2)² to compensate for the extra term you introduced.
3. Step 3: Add c to get the final expression: (x + b/2)² − (b/2)² + c.

When a ≠ 1, factor out a from the first two terms before completing the square inside the bracket. This is a common Grade 8/9 extension.

Worked Examples

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