The Quadratic Formula: GCSE Maths Guide with Worked Examples

The quadratic formula is a powerful tool in GCSE Maths that lets you solve any quadratic equation, even when factorising is difficult or impossible. This guide is part of our GCSE Maths revision guide — start there for an overview of all topics. It is a Higher tier topic, and questions on it appear regularly in exams. This guide walks you through the formula, when to use it, worked examples, and the most common pitfalls.

The Formula

For a quadratic equation in the form ax² + bx + c = 0, the solutions are:

x = (-b +/- square root of (b² - 4ac)) / 2a

The ”+/-” symbol means there are usually two solutions: one where you add the square root and one where you subtract it. These two values of x are called the roots of the equation.

When to Use It

There are three main methods for solving quadratic equations at GCSE:

1. Factorising — the quickest method when it works, but only works for “nice” numbers
2. Completing the square — useful for finding the turning point of a quadratic graph
3. The quadratic formula — works for every quadratic equation, including those with awkward or irrational roots

Use the quadratic formula when:

• The equation does not factorise neatly
• The question specifically asks you to use the formula
• You need to give answers to a certain number of decimal places or significant figures (a strong hint that the roots are not whole numbers)

Step-by-Step Method

1. Rearrange the equation into the form ax² + bx + c = 0 (everything on one side, zero on the other)
2. Identify the values of a, b, and c
3. Substitute into the formula
4. Calculate the discriminant (b² - 4ac) first
5. Find the two solutions using + and - separately
6. Round your answers if the question asks for a specific degree of accuracy

Worked Examples

Example 1: Straightforward Application

Solve 2x² + 5x - 3 = 0, giving answers to 2 decimal places.

Identify: a = 2, b = 5, c = -3

Discriminant: b² - 4ac = 25 - 4(2)(-3) = 25 + 24 = 49

Substitute:

x = (-5 +/- square root of 49) / (2 x 2)

x = (-5 +/- 7) / 4

Two solutions:

x = (-5 + 7) / 4 = 2/4 = 0.50

x = (-5 - 7) / 4 = -12/4 = -3.00

In this case the roots are rational, so factorising would also have worked. But the formula gives you the answer regardless.

Example 2: Irrational Roots

Solve x² - 6x + 2 = 0, giving answers to 3 significant figures.

Identify: a = 1, b = -6, c = 2

Discriminant: b² - 4ac = 36 - 4(1)(2) = 36 - 8 = 28

Substitute:

x = (6 +/- square root of 28) / 2

square root of 28 = 5.2915…

x = (6 + 5.2915…) / 2 = 11.2915… / 2 = 5.65 (3 s.f.)

x = (6 - 5.2915…) / 2 = 0.7085… / 2 = 0.354 (3 s.f.)

Notice how b = -6, so -b = 6. Getting the sign right here is critical.

Example 3: Rearranging First

Solve 3x² = 7x - 1, giving answers to 2 decimal places.

First, rearrange: 3x² - 7x + 1 = 0

Identify: a = 3, b = -7, c = 1

Discriminant: b² - 4ac = 49 - 4(3)(1) = 49 - 12 = 37

Substitute:

x = (7 +/- square root of 37) / 6

square root of 37 = 6.0828…

x = (7 + 6.0828…) / 6 = 13.0828… / 6 = 2.18 (2 d.p.)

x = (7 - 6.0828…) / 6 = 0.9172… / 6 = 0.15 (2 d.p.)

The Discriminant

The expression under the square root sign, b² - 4ac, is called the discriminant. It tells you how many real roots the equation has:

• b² - 4ac > 0: two distinct real roots (the parabola crosses the x-axis twice)
• b² - 4ac = 0: one repeated root (the parabola just touches the x-axis)
• b² - 4ac < 0: no real roots (the parabola does not cross the x-axis)

Why Does This Matter?

Exam questions sometimes ask you to “show that the equation has no real roots” or “find the values of k for which the equation has equal roots.” In these cases, you calculate the discriminant and set it greater than, equal to, or less than zero as required.

Example: Using the Discriminant

Show that x² + 2x + 5 = 0 has no real roots.

b² - 4ac = 4 - 4(1)(5) = 4 - 20 = -16

Since the discriminant is negative (-16 < 0), there are no real roots.

Connection to Quadratic Graphs

The roots of a quadratic equation are the x-coordinates where the parabola crosses the x-axis. If you sketch y = ax² + bx + c:

• Two roots means the graph crosses the x-axis at two points
• One repeated root means the graph touches the x-axis at its turning point
• No real roots means the graph sits entirely above (or entirely below) the x-axis

The quadratic formula does not just give you numbers to write down. It tells you about the shape and position of the graph.

Common Mistakes to Avoid

Sign Errors with Negative b

When b is negative, -b becomes positive. This is the number one source of errors. For example, if b = -4, then -b = 4, and b² = 16 (not -16). Write out each step carefully.

Forgetting the +/- Sign

The formula always gives two solutions (unless the discriminant is zero). If you only calculate one answer, you have lost marks. Always compute both the + and - versions.

Not Setting the Equation to Zero First

The formula only works when the equation is in the form ax² + bx + c = 0. If the equation is given as x² + 3x = 10, you must rearrange to x² + 3x - 10 = 0 before identifying a, b, and c. Getting c wrong because you did not rearrange is a very common error.

Dividing by 2a, Not Just 2

The denominator is 2a, not 2. If a = 3, you divide the entire numerator by 6, not by 2. Students who write the formula from memory sometimes miss this detail.

Premature Rounding

Do not round the discriminant or the square root before calculating the final answer. Keep all decimal places on your calculator until the very last step, then round to the required accuracy. Rounding too early introduces errors that compound through the calculation.

Exam Technique Tips

Write the formula out. Before substituting, write the general formula. This earns a method mark and helps you organise your working.

Calculate the discriminant separately. Work out b² - 4ac as a standalone step. This reduces errors and makes your working clearer.

Give both solutions. Unless the question only asks for positive values or the discriminant is zero, always present two answers.

Check by substitution. If time allows, substitute your answers back into the original equation to verify they work. This is especially useful for catching sign errors.

Summary

The quadratic formula solves any equation of the form ax² + bx + c = 0. Identify a, b, and c carefully, watch your signs, calculate the discriminant first, and always give two solutions. The discriminant tells you how many roots exist before you even solve.

Related GCSE Maths Guides

• Algebra revision guide — from simplifying expressions to solving quadratics by factorising
• GCSE Maths formula sheet guide — which formulas are provided in the exam and which you must memorise
• GCSE Further Maths revision guide — extends algebra into calculus, proof, and complex numbers

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