The Quadratic Formula: When and How to Use It in IGCSE Maths

What Is the Quadratic Formula?

The quadratic formula solves any equation in the form ax² + bx + c = 0. It states that x = (−b ± √(b² − 4ac)) / 2a. While factorising and completing the square are alternative methods, the quadratic formula works for every quadratic equation, including those with irrational or complex roots. This makes it the go-to method when other techniques fail or when the question specifically instructs you to use it.

At the IGCSE level, Cambridge provides the quadratic formula on the formula sheet for Paper 2 and Paper 4. However, you still need to know how to identify a, b, and c correctly, substitute them without error, and simplify the result.

When Should You Use the Quadratic Formula?

There are specific situations where the quadratic formula is the best or only practical choice:

• The equation does not factorise neatly. If you cannot find two numbers that multiply to give ac and add to give b, factorising will not work. The quadratic formula handles these cases effortlessly.
• The question asks you to give answers to a specific number of decimal places or significant figures. This is a strong hint that the roots are irrational, and the formula is the intended method.
• The question says “use the formula” or “solve, giving your answers correct to 2 decimal places.” These instructions explicitly point you towards the quadratic formula.
• You are solving a problem that produces a quadratic after rearranging. In context questions involving area, projectile motion, or simultaneous equations, the resulting quadratic often does not factorise.

On the other hand, if the equation factorises easily, factorising is faster and earns the same marks. Always check whether simple factorisation works before reaching for the formula.

Identifying a, b, and c

The most critical step is correctly identifying the coefficients. The equation must be in the standard form ax² + bx + c = 0 before you read off a, b, and c.

Consider the equation 3x² − 7x + 2 = 0. Here, a = 3, b = −7, and c = 2. Note that b is negative because the term is −7x.

Now consider 5x² + 3 = 8x. Before identifying coefficients, rearrange to standard form: 5x² − 8x + 3 = 0. Now a = 5, b = −8, and c = 3. Failing to rearrange first is one of the most common errors students make.

Another tricky case: x² − 5 = 0. This is still quadratic, with a = 1, b = 0, and c = −5.

Step-by-Step Application

Let us solve 2x² + 5x − 3 = 0 using the quadratic formula.

First, identify the coefficients: a = 2, b = 5, c = −3.

Next, calculate the discriminant: b² − 4ac = 25 − 4(2)(−3) = 25 + 24 = 49.

Now substitute into the formula: x = (−5 ± √49) / (2 × 2) = (−5 ± 7) / 4.

This gives two solutions: x = (−5 + 7)/4 = 2/4 = 0.5, and x = (−5 − 7)/4 = −12/4 = −3.

So x = 0.5 or x = −3.

Notice that the discriminant was a perfect square (49), which means this equation could also have been factorised. When the discriminant is not a perfect square, you know the roots will be irrational, confirming that the formula was the right approach.

The Discriminant: A Powerful Diagnostic Tool

The expression b² − 4ac, known as the discriminant, tells you about the nature of the roots before you even solve the equation:

• b² − 4ac > 0: Two distinct real roots. The equation crosses the x-axis at two points.
• b² − 4ac = 0: One repeated root (the parabola touches the x-axis at exactly one point).
• b² − 4ac < 0: No real roots. The parabola does not cross the x-axis.

Cambridge sometimes asks you to find the value of a constant for which a quadratic has equal roots or no real roots. Setting the discriminant equal to zero or applying an inequality is the key technique here.

For example, if x² + kx + 9 = 0 has equal roots, then k² − 36 = 0, giving k = 6 or k = −6.

Common Mistakes to Avoid

Years of examining student work reveal the same recurring errors:

• Forgetting the negative sign on b. If b = −7, then −b = 7. Many students accidentally write −(−7) as −7 instead of +7.
• Errors in calculating b² − 4ac. Be especially careful when c is negative, as −4a(−c) becomes positive. Write out each step.
• Dividing only part of the numerator by 2a. The entire expression (−b ± √(b² − 4ac)) must be divided by 2a, not just the square root term.
• Forgetting the ± sign. There are always two possible values (unless the discriminant is zero). Dropping the minus option means losing one solution.
• Not rearranging to standard form first. If the equation is not equal to zero, the values of a, b, and c will be wrong.

Practising for Exam Success

To build confidence with the quadratic formula, follow this progression:

• Start with equations where the discriminant is a perfect square so you can verify by factorising
• Move to equations requiring decimal answers and practise rounding correctly
• Tackle word problems that require setting up a quadratic equation first
• Time yourself on past paper questions to build speed and accuracy

Write out every step clearly in your working. Examiners award method marks for correct substitution into the formula, even if your final answer contains an arithmetic error. Showing your working is therefore essential for maximising your marks.

Linking to Other Topics

The quadratic formula connects to many other IGCSE topics. You will use it when solving simultaneous equations where one equation is quadratic. It appears in coordinate geometry when finding intersection points of lines and curves. It is also relevant in problems involving areas, volumes, and optimisation where the resulting equation is quadratic.

Understanding when and how to deploy the formula across different contexts is what separates students who score well from those who struggle.

Get Expert Help with IGCSE Maths

Struggling with quadratic equations or unsure when to use which method? Our IGCSE Maths tutors provide step-by-step guidance tailored to your learning style, helping you build confidence and accuracy for exam day.

Book a Free Trial Class | WhatsApp Us