Quadratic equations are a core part of the GCSE Maths curriculum across all exam boards (AQA, Edexcel, OCR). If you've ever seen an equation like (x² - 5x + 6 = 0) and wondered how to find the values of (x), you're in the right place. This guide will walk you through the three main methods you need to know, helping you decide which one to use and how to apply it correctly.
What is a Quadratic Equation?
A quadratic equation is any equation that can be rearranged into the standard form:
ax² + bx + c = 0
Where (a), (b), and (c) are numbers (constants), and (a ≠ 0). The solutions to a quadratic equation are called the roots or x-intercepts. They are the points where the graph of the equation (a parabola) crosses the x-axis. A quadratic can have 2, 1, or 0 real solutions.
The Three Key Methods for Solving Quadratics
For your GCSE, you need to be confident with these three techniques. The best method often depends on the specific equation you're given.
flowchart TD
A["Start: ax² + bx + c = 0"] --> B["Can it be factorised easily?"]
B -- Yes --> C["Use Factorising"]
B -- No --> D["a = 1 and b is even?"]
D -- Yes --> E["Consider Completing the Square"]
D -- No --> F["Use the Quadratic Formula"]
C --> G["Found Solutions"]
E --> G
F --> G
1. Factorising
This is often the quickest method, but it only works nicely for certain quadratics.
When to use it: Look for simple integer factors. It works best when (a = 1) or when common factors are obvious.
Step-by-Step for (x² + bx + c = 0):
Set up two sets of brackets: ((x \quad )(x \quad ) = 0).
Find two numbers that multiply to give (c) and add to give (b).
Place these numbers in the brackets.
- Set each bracket equal to zero and solve for (x).
Example: Solve (x² - 5x + 6 = 0)
We need two numbers that multiply to make (+6) and add to make (-5).
Those numbers are (-2) and (-3) (because (-2 × -3 = +6) and (-2 + -3 = -5)).
- So, ((x - 2)(x -
- 3) = 0).
- If ((x -
- 2) = 0), then (x = 2).
- If ((x -
- 3) = 0), then (x = 3).
- Solutions: (x = 2) and (x = 3).
2. The Quadratic Formula
This is your most powerful tool—it works for every quadratic equation.
When to use it: When factorising is tricky or impossible (especially when the roots are decimals or fractions).
The Formula:
x = -b ± √b^2 - 4ac2a
Step-by-Step:
Ensure your equation is in the form (ax² + bx + c = 0) and identify (a), (b), and (c).
Substitute these values carefully into the formula.
Calculate the discriminant ((b² - 4ac)) first. This tells you about the roots:
(b² - 4ac > 0): Two distinct real roots.
(b² - 4ac = 0): One repeated real root.
(b² - 4ac < 0): No real roots (not on GCSE Foundation).
- Work through the calculation to find your solutions.
Example: Solve (2x² + 3x - 5 = 0)
Here, (a = 2), (b = 3), (c = -5).
Substitute: x = -3 ± √3^2 - 4 × 2 × (-52 × 2)
Simplify: (x = -3 ± √9 + 404 = -3 ± √494 = -3 ± 74)
Calculate both solutions:
(x = -3 + 74 = 44 = 1)
(x = -3 - 74 = -104 = -2.5)
- Solutions: (x = 1) and (x = -2.5).
3. Completing the Square
This method is essential for higher-tier GCSE as it helps with sketching graphs and solving harder problems.
When to use it: When asked specifically, or when solving equations of the form (x² + bx + c = 0) where (b) is even. It's also key for deriving the quadratic formula!
Step-by-Step for (x² + bx + c = 0):
Focus on the (x² + bx) part.
Rewrite it as ((x + b2)² - (b2)²).
Bring the constant term ((c)) into the expression.
- Rearrange the equation to solve for (x).
Example: Solve (x² + 6x + 2 = 0) by completing the square.
(b = 6), so (b2 = 3).
Write: (x² + 6x) as ((x + 3)² - 3² = (x+3)² - 9).
The full equation becomes: ((x+3)² - 9 + 2 = 0), which simplifies to ((x+3)² - 7 = 0).
Rearrange: ((x+3)² = 7).
Square root both sides: (x + 3 = ±√7).
Solve: (x = -3 ±√7).
- Solutions: (x = -3 + √7) and (x = -3 - √7) (you can leave these in surd form).
Practice Makes Perfect
The key to mastering quadratics is practice. Try these steps with past paper questions:
Identify the form. Is it already equal to zero? What are (a), (b), and (c)?
Choose your method. Use the flowchart as a guide.
Work neatly. Avoid arithmetic errors, especially with signs in the quadratic formula.
- Check your answers. Substitute your solutions back into the original equation to verify they work.
Common GCSE Exam Pitfalls to Avoid
Forgetting the (±) sign when square rooting in completing the square or the quadratic formula. This often means missing one of the two solutions.
Incorrectly identifying (a), (b), and (c) in the quadratic formula, especially if the equation is not in the standard form (e.g., (3x² = 5x + 2)). Always rearrange to (ax2+bx+c (so this becomes (3x² - 5x - 2 = 0)).
Trying to factorise everything. Some quadratics don't factorise neatly—know when to switch to the formula.
- Arithmetic errors when calculating the discriminant. Take your time.
- By understanding these three methods and when to apply them, you'll be well-equipped to tackle any quadratic equation that comes up in your GCSE Maths exam. Good luck with your revision!