How to Solve Quadratic Equations in GCSE Maths

QQuadratic equations might look intimidating at first, but they follow clear patterns that become straightforward once you know the steps. Whether you’re aiming for grade 7, 8, or 9, mastering quadratics will boost your confidence and exam performance significantly.

This guide shows you exactly how to solve any quadratic equation using three reliable methods, with crystal-clear examples you can actually follow.

What Are Quadratic Equations?

A quadratic equation is any equation with x² as the highest power. They always follow this pattern:

ax² + bx + c = 0

Where a, b, and c are just numbers.

Common examples:

• x² + 5x + 6 = 0
• 2x² – 7x + 3 = 0
• x² – 9 = 0

Why they matter for GCSE:

• Worth lots of marks across all exam papers
• Appear in graph questions and problem-solving
• Essential for A Level preparation
• Often combined with other topics for high-mark questions

Method 1: Solving by Factorising (Usually Quickest)

When you can factorise the quadratic, this method is fastest and most reliable.

Basic Factorising Steps

Step 1: Make sure the equation equals zero Step 2: Factorise the left side into two brackets Step 3: Set each bracket equal to zero Step 4: Solve each simple equation

Worked Example 1

Solve x² + 5x + 6 = 0

Step 1: Already equals zero ✓

Step 2: Factorise x² + 5x + 6 We need two numbers that:

• Multiply to give 6
• Add to give 5

Let’s try: 2 × 3 = 6 and 2 + 3 = 5 ✓

So x² + 5x + 6 = (x + 2)(x + 3)

Step 3: Set each bracket equal to zero (x + 2)(x + 3) = 0

This means either: x + 2 = 0 OR x + 3 = 0

Step 4: Solve each equation From x + 2 = 0: x = -2 From x + 3 = 0: x = -3

Final Answer: x = -2 or x = -3

Worked Example 2

Solve x² – x – 12 = 0

Step 1: Already equals zero ✓

Step 2: Factorise x² – x – 12 We need two numbers that:

• Multiply to give -12
• Add to give -1

Since we need -12, one number is positive and one is negative.

Let’s try: 3 × (-4) = -12 and 3 + (-4) = -1 ✓

So x² – x – 12 = (x + 3)(x – 4)

Step 3: Set each bracket equal to zero x + 3 = 0 OR x – 4 = 0

Step 4: Solve each equation From x + 3 = 0: x = -3 From x – 4 = 0: x = 4

Final Answer: x = -3 or x = 4

Method 2: The Quadratic Formula (Always Works)

When factorising looks tricky, the quadratic formula is your reliable backup. Yes, it looks intimidating at first, but it’s actually just a recipe to follow step by step.

The Formula (Don’t Panic!)

x = (-b ± √(b² – 4ac)) / 2a

What do the letters mean? For any quadratic in the form ax² + bx + c = 0:

• a = the number in front of x²
• b = the number in front of x
• c = the number on its own

The ± symbol means you’ll get TWO answers – one using + and one using -.

Step-by-Step Method

Step 1: Make sure your equation equals zero Step 2: Identify your a, b, and c values Step 3: Substitute carefully into the formula
Step 4: Use your calculator to find both solutions

Worked Example 3 (Simple Steps)

Solve x² – 4x – 1 = 0

Step 1: Already equals zero ✓

Step 2: Find a, b, and c

• a = 1
• b = -4
• c = -1

Step 3: Use the formula x = (-b ± √(b² – 4ac)) / 2a

Step 4: Work it out bit by bit

• -b = 4
• b² = 16
• 4ac = -4
• So: 16 – (-4) = 20
• 2a = 2

Step 5: Put it together x = (4 ± √20) / 2

Step 6: Get your answers √20 = 4.47 (using calculator)

Answer 1: x = (4 + 4.47) / 2 = 4.24 Answer 2: x = (4 – 4.47) / 2 = -0.24

Final Answer: x = 4.24 or x = -0.24

Worked Example 4 (When a ≠ 1)

Solve 2x² + 3x – 1 = 0

Step 1: Find a, b, and c

• a = 2
• b = 3
• c = -1

Step 2: Work out each bit

• -b = -3
• b² = 9
• 4ac = -8
• So: 9 – (-8) = 17
• 2a = 4

Step 3: Put into formula x = (-3 ± √17) / 4

Step 4: Use calculator √17 = 4.12

Answer 1: x = (-3 + 4.12) / 4 = 0.28 Answer 2: x = (-3 – 4.12) / 4 = -1.78

Final Answer: x = 0.28 or x = -1.78

Calculator Tips That Work

Avoid errors:

• Use brackets: (-3)² not -3²
• Check b² – 4ac is positive first
• Do one step at a time

Quick check: Put your answer back into the original equation!

Method 3: Completing the Square (For Exact Answers)

This method might seem complex at first, but it’s actually quite logical once you see the pattern. It’s particularly useful when you need exact answers with surds.

The Step-by-Step Process

Step 1: Make sure the coefficient of x² is 1 Step 2: Take half the coefficient of x, then square it Step 3: Add and subtract this number Step 4: Rearrange into perfect square form Step 5: Solve the resulting equation

Worked Example 5 (Simple Steps)

Solve x² + 6x + 2 = 0

Step 1: Half the x number and square it x has 6 in front Half of 6 = 3 3² = 9

Step 2: Add and subtract 9 x² + 6x + 2 = 0 x² + 6x + 9 – 9 + 2 = 0 x² + 6x + 9 – 7 = 0

Step 3: Make the perfect square x² + 6x + 9 = (x + 3)²

So: (x + 3)² – 7 = 0 Therefore: (x + 3)² = 7

Step 4: Solve x + 3 = ±√7 x = -3 ± √7

Step 5: Get decimal answers √7 = 2.65

Answer 1: x = -3 + 2.65 = -0.35 Answer 2: x = -3 – 2.65 = -5.65

Final Answer: x = -0.35 or x = -5.65

When to Use Each Method

Use Factorising When:

• Numbers look “nice” (small integers)
• You can easily spot factor pairs
• Working without a calculator

Use the Quadratic Formula When:

• Factorising looks difficult
• You need decimal answers
• The numbers are messy fractions

Use Completing the Square When:

• You need exact answers with surds
• Question specifically asks for this method
• Working with graphs (finding turning points)

Common Mistakes and How to Avoid Them

Mistake 1: Forgetting the ± in the Formula

Wrong: x = (-b + √(b² – 4ac)) / 2a ← Only using +

Right: x = (-b ± √(b² – 4ac)) / 2a ← Using both + and –

Remember: You almost always get TWO answers

Mistake 2: Sign Errors

Watch out for:

• If b = -4, then -b = +4
• If c = -1, then 4ac is negative

Pro tip: Use brackets: (-4)² not -4²

Mistake 3: Calculator Errors

Common problems:

• Forgetting brackets
• Not finding both answers
• Rounding too early

Better approach:

• Work out b² – 4ac first
• Use brackets for everything
• Find both solutions separately

Mistake 4: Not Checking Answers

Always check: Put your answer back into the original equation.

Example: If x = 2 for x² – 5x + 6 = 0: 2² – 5(2) + 6 = 4 – 10 + 6 = 0 ✓

If it doesn’t equal zero, you made an error.

Practice Questions

Easy Level:

1. x² + 7x + 12 = 0
2. x² – 5x + 6 = 0
3. x² – 9 = 0

Medium Level: 4. x² + x – 6 = 0 5. x² – 6x + 5 = 0 6. 2x² + 5x + 2 = 0

Challenge Level: 7. x² + 4x – 3 = 0 8. 3x² – 7x + 2 = 0 9. x² – 2x – 4 = 0

Solutions:

1. x = -3 or x = -4
2. x = 2 or x = 3
3. x = 3 or x = -3
4. x = 2 or x = -3

(Try the rest yourself and check with your teacher!)

How Expert Tutoring Helps with Quadratics

Many students find quadratic equations challenging because they involve multiple steps and method selection. Working with an experienced tutor can help you build confidence much faster.

Personalised Learning That Builds Confidence

Ramsay’s Expertise – Oxford Mathematics Graduate: Ramsay achieved a Distinction in Prelims at Oxford and was awarded both the East Scholarship and the Fitzgerald Prize for academic excellence. He has a strong record in mathematical competitions, achieving Distinctions in both rounds of the British Mathematical Olympiad.

Ramsay’s teaching style is clear, calm, and intuitive, helping students to build both technical fluency and confidence in problem-solving. He is especially effective with high-aspiring students and those preparing for competitive university applications.

Real Student Success Stories

Maria’s Proven Results: Cambridge Mathematics graduate Maria helped one student progress from Grade D to A* in A Level Maths in just three months. Being the first person in her school’s history to be admitted into Oxbridge for Mathematics, Maria knows exactly how to prepare students for mathematical success.

Jonathan’s Transformation: From the actual feedback we received: “Noah’s good to talk to, always asking how my week was and what I did. Then, in the lesson, Noah makes sure I understand before moving onto other topics.”

Students consistently report feeling more confident after lessons, with parents noting significant improvements in their children’s mathematical understanding and exam performance.

What Makes Our Approach Different

Individual Assessment:

• Identify where students get confused with algebra
• Address specific gaps in equation-solving skills
• Build systematic checking methods

Method Selection Training:

• Learn when to use each method
• Practice choosing the best approach quickly
• Develop confidence with formula work

Exam Confidence:

• Practice with real GCSE question styles
• Learn time management for different methods
• Build resilience when facing difficult problems

Building Exam Success

Time Management Tips

For non-calculator papers:

• Try factorising first (it’s usually quicker)
• Don’t spend too long if factorising isn’t obvious
• Use completing the square for exact answers

For calculator papers:

• Use the quadratic formula for messy numbers
• Always check your answers by substituting back
• Round appropriately (usually 2 or 3 decimal places)

Common Exam Question Types

Direct solving: “Solve x² + 5x – 6 = 0” Method specification: “Solve by completing the square” Applied problems: “A rectangle has area 12 cm². Its length is 1 cm more than its width. Find the dimensions.” Graph connections: “Find where y = x² – 3x – 4 crosses the x-axis”

Ready to Master Quadratic Equations?

Quadratic equations become much more manageable when you understand the step-by-step methods and know which one to choose. The key is building confidence through clear explanations and plenty of practice.

If you’re finding quadratics challenging, you’re not alone. Many successful students initially struggle with the algebra and method selection. The difference is getting clear, patient explanation when you need it.

How Expert Tutoring Transforms Quadratic Understanding

Many students find quadratic equations initially confusing because they involve multiple steps and different methods. Working with an experienced maths tutor can dramatically accelerate your understanding and confidence.

Real Student Success

Sarah’s Breakthrough: After struggling with sign errors and factorising, Sarah worked with Maria (Cambridge Mathematics) for just six weeks before her mocks.

“Maria taught me systematic checking methods that eliminated my silly mistakes. I went from dreading quadratic questions to seeing them as easy marks.”

Ramsay’s Approach – Oxford Mathematics Graduate: Ramsay specialises in helping students see the logical patterns in quadratic equations rather than memorising disconnected procedures.

“Most students get overwhelmed by the three different methods. I help them understand when and why to use each approach, so they feel confident choosing the right method in exam conditions.”

James’s Transformation: Working with our GCSE specialist Morgan (Oxford Chemistry, First Class Honours), James went from consistently making algebraic errors to achieving full marks on quadratic questions.

“Morgan showed me that quadratics aren’t about remembering formulas – they’re about recognising patterns. Once I understood the logic, everything clicked into place.”

How We Can Help

Our GCSE Mathematics tutors specialise in:

• Systematic algebraic skill building to eliminate common errors
• Method selection strategies for different question types
• Confidence development through structured practice
• Exam technique optimisation to maximise marks

Success stories speak for themselves: “I thought I was hopeless at algebra until I started working with Greenhill. Now quadratics are actually my favourite topic!” – Emma, Grade 8 achieved

“The systematic approach my tutor taught me eliminated all my silly mistakes. I gained 15 marks just on quadratic questions.” – Tom, Grade 9 achieved

Experience the Greenhill Academics Difference

Our students consistently exceed their predicted grades because we believe every learner can achieve mathematical success with the right support.

Ready to discover your mathematical potential?

Book your free consultation today and experience how our Oxbridge-educated tutors make mathematics accessible, engaging, and achievable.

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