GCSE Maths Revision Guide: Quadratic Equations and Graphs

Quadratics are one of the biggest topics in GCSE Maths. They show up in algebra questions, graphs, worded problems and even in some physics style questions about throwing a ball in the air.

If you can handle quadratic equations and graphs with confidence, you unlock a lot of easy and medium marks across the paper.

This guide will walk you through:

• What a quadratic is
• The different forms of quadratic equations
• How to solve quadratics by factorising
• How to solve using the quadratic formula
• Completing the square (mainly Higher tier)
• Sketching and interpreting quadratic graphs
• Links to real exam style questions and revision tips

Everything here is in UK GCSE language and aimed at both Foundation and Higher students. Higher tier will need to know all of it in more depth.

1. What is a quadratic?

A quadratic is any expression or equation where the highest power of the variable is 2.

Common examples:

• (x^2 + 5x + 6)
• (2x^2 - 7x + 3)
• (x^2 - 9)

These are quadratic expressions. When you set them equal to something, usually zero, you get a quadratic equation.

Example:

• (x^2 + 5x + 6 = 0)

The graph of a quadratic is a parabola. It can either be:

• Smiling (a U shape that opens upwards) when the coefficient of (x^2) is positive
• Frowning (n shape that opens downwards) when the coefficient of (x^2) is negative

At GCSE, most of your quadratics will be smiling parabolas.

2. Forms of a quadratic

You will usually see quadratics written in three main forms. Being able to recognise these makes everything else easier.

2.1 Standard form

This is the one you see most often:

[ ax^2 + bx + c = 0 ]

Where:

• (a), (b), (c) are numbers
• (a \ne 0)

Example: (2x^2 - 3x - 5 = 0)

This is the form you use for the quadratic formula.

2.2 Factorised form

This is where the quadratic is written as a product of brackets.

Example:

[ x^2 + 5x + 6 = (x + 2)(x + 3) ]

This form is really useful because you can read off the solutions straight away.

If ((x + 2)(x + 3) = 0), then either:

• (x + 2 = 0) so (x = -2)
• (x + 3 = 0) so (x = -3)

So the solutions are (x = -2) and (x = -3).

2.3 Completed square form (Higher)

This form looks like:

[ a(x + p)^2 + q ]

For example:

[ x^2 + 4x + 1 = (x + 2)^2 - 3 ]

This form is brilliant for sketching graphs, because you can read off the turning point straight away. More on that later.

3. Solving quadratic equations by factorising

For Foundation and Higher, this is the first method you should be confident with.

3.1 Simple quadratics where (a = 1)

These are quadratics like (x^2 + 7x + 12 = 0).

Step 1: Look for two numbers that multiply to give (c) and add to give (b).

For (x^2 + 7x + 12):

• You need two numbers that multiply to 12 and add to 7.
• The pair 3 and 4 works, because 3 × 4 = 12 and 3 + 4 = 7.

So:

[ x^2 + 7x + 12 = (x + 3)(x + 4) ]

Now set equal to zero:

[ (x + 3)(x + 4) = 0 ]

So the solutions are (x = -3) and (x = -4).

Exam tip: Once you have your answers, quickly substitute one back into the original equation to check it works.

3.2 Quadratics where (a \ne 1)

Example: (2x^2 + 7x + 3 = 0).

You need two numbers that multiply to (a \times c) and add to (b).

Here:

• (a = 2), (b = 7), (c = 3)
• (a \times c = 2 \times 3 = 6)
• You need two numbers that multiply to 6 and add to 7. That is 6 and 1.

Rewrite the middle term using these numbers:

[ 2x^2 + 6x + x + 3 = 0 ]

Now factorise in pairs:

[ 2x(x + 3) + 1(x + 3) = 0 ]

Factorise the common bracket:

[ (2x + 1)(x + 3) = 0 ]

So the solutions are:

• (2x + 1 = 0 \Rightarrow x = -\tfrac{1}{2})
• (x + 3 = 0 \Rightarrow x = -3)

3.3 Difference of two squares

This is a special pattern you should spot instantly:

[ x^2 - 9 = (x - 3)(x + 3) ]

In general:

[ a^2 - b^2 = (a - b)(a + b) ]

Example:

Solve (x^2 - 16 = 0).

Factorise:

[ (x - 4)(x + 4) = 0 ]

Solutions: (x = 4) or (x = -4).

Common mistake: Trying to factorise things like (x^2 + 9) as ((x + 3)(x + 3)). That is wrong. (x^2 + 9) does not factorise nicely over the reals.

4. Solving quadratics using the quadratic formula

Sometimes the quadratic will not factorise nicely. That is where the quadratic formula comes in.

For (ax^2 + bx + c = 0):

[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

The expression under the square root, (b^2 - 4ac), is called the discriminant.

• If (b^2 - 4ac > 0) you get two distinct real solutions.
• If (b^2 - 4ac = 0) you get one repeated solution (the graph just touches the x axis).
• If (b^2 - 4ac < 0) there are no real solutions (graph does not cross the x axis).

4.1 Example question

Solve (2x^2 - 3x - 5 = 0).

Here:

• (a = 2)
• (b = -3)
• (c = -5)

Work out the discriminant:

[ b^2 - 4ac = (-3)^2 - 4(2)(-5) = 9 + 40 = 49 ]

Now use the formula:

[ x = \frac{-(-3) \pm \sqrt{49}}{2 \cdot 2} = \frac{3 \pm 7}{4} ]

So:

• (x = \frac{3 + 7}{4} = \frac{10}{4} = 2.5)
• (x = \frac{3 - 7}{4} = \frac{-4}{4} = -1)

Always simplify your answers where possible. Depending on the question, decimals or fractions might be more appropriate.

Exam tip: If the question is non calculator and the numbers look nasty, check whether you were meant to factorise instead.

5. Completing the square (Higher)

Completing the square is a Higher tier skill. It lets you:

• Rewrite (ax^2 + bx + c) in completed square form
• Find the turning point of the graph
• Solve certain equations more neatly

5.1 Basic pattern

Start with something like (x^2 + 6x + 5).

Focus on the (x^2 + 6x) part. Half the coefficient of x and square it.

• Half of 6 is 3
• 3 squared is 9

So write:

[ x^2 + 6x + 9 - 9 + 5 ]

Group the first three terms:

[ (x^2 + 6x + 9) - 4 ]

Now the bracket is a perfect square:

[ (x + 3)^2 - 4 ]

So (x^2 + 6x + 5 = (x + 3)^2 - 4).

5.2 Turning points from completed square form

For (y = (x + p)^2 + q):

• The turning point is at ((-p, q))
• The graph is a translation of (y = x^2)

Example:

If (y = (x - 2)^2 + 5), the turning point is at ((2, 5)).

If (y = (x + 3)^2 - 4), the turning point is at ((-3, -4)).

This is much quicker than using calculus at GCSE.

6. Quadratic graphs

6.1 Basic shape

The simplest quadratic graph is (y = x^2).

• It is a U shaped curve with its minimum at the origin ((0, 0))
• It is symmetrical about the y axis

If you sketch (y = x^2), then changing the equation transforms the graph.

6.2 Effect of (a) in (y = ax^2)

• If (a > 0), the graph opens upwards
• If (a < 0), the graph opens downwards
• If (|a| > 1), the graph is narrower (steeper)
• If (|a| < 1), the graph is wider (flatter)

Example:

• (y = 2x^2) is narrower than (y = x^2)
• (y = 0.5x^2) is wider than (y = x^2)

6.3 From equation to graph (standard form)

For (y = ax^2 + bx + c):

• (c) is the y intercept (where the graph crosses the y axis)
• The x intercepts (where the graph crosses the x axis) are the solutions to (ax^2 + bx + c = 0)
• The turning point can be found by completing the square or using the formula (x = -\tfrac{b}{2a}) then substituting back in

Example:

Consider (y = x^2 - 4x + 3).

• y intercept is (c = 3)
• Solve (x^2 - 4x + 3 = 0) by factorising: ((x - 1)(x - 3) = 0), so x intercepts are 1 and 3
• Turning point: the x coordinate is (x = -\tfrac{b}{2a} = -\tfrac{-4}{2 \cdot 1} = 2). Substitute back: (y = 2^2 - 4(2) + 3 = 4 - 8 + 3 = -1). So turning point is at ((2, -1)).

Sketching this, you would mark the intercepts at (1, 0), (3, 0), the turning point at (2, -1), and join with a smooth U shaped curve.

6.4 Graphs and inequalities

Exam boards like to mix inequalities with quadratics.

Example question:

> The graph of (y = x^2 - 4x + 3) is drawn. Use the graph to solve (x^2 - 4x + 3 \le 0).

Because (x^2 - 4x + 3 \le 0) means the graph is on or below the x axis, you look at where the curve is below the axis.

From the intercepts earlier, the curve crosses at x = 1 and x = 3 and it is below the axis between those points.

So the solution is:

[ 1 \le x \le 3 ]

7. Quadratics in context questions

Quadratics often appear in real world or exam style context questions.

7.1 Area problems

A common style is rectangles and fencing.

Example:

> A rectangular garden is x metres wide and (x + 4) metres long. The area is 96 square metres. Form and solve a quadratic equation to find the value of x.

Area of a rectangle is width multiplied by length:

[ x(x + 4) = 96 ]

Expand:

[ x^2 + 4x = 96 ]

Bring all terms to one side:

[ x^2 + 4x - 96 = 0 ]

Now solve this quadratic. Look for two numbers that multiply to -96 and add to 4.

12 and -8 work because 12 × (-8) = -96 and 12 + (-8) = 4.

So:

[ x^2 + 4x - 96 = (x + 12)(x - 8) = 0 ]

Solutions:

• (x = -12) or (x = 8)

In context, a negative width makes no sense, so (x = 8) metres.

7.2 Height time graphs (projectiles)

Higher tier in particular like the classic ball thrown in the air question.

Example:

> The height h metres of a ball above the ground t seconds after it is thrown is given by (h = -5t^2 + 20t + 1).
>
> a) Find the height of the ball when t = 2.
>
> b) After how many seconds does the ball hit the ground?

For part a, substitute t = 2:

[ h = -5(2^2) + 20(2) + 1 = -5(4) + 40 + 1 = -20 + 40 + 1 = 21 ]

So the height is 21 metres.

For part b, hitting the ground means (h = 0):

[ -5t^2 + 20t + 1 = 0 ]

This will not factorise nicely, so use the quadratic formula with:

• (a = -5)
• (b = 20)
• (c = 1)

You will get two values for t. One will be negative (not realistic here) and one positive. The positive one is the time when the ball hits the ground.

In an exam, give your answer to a sensible number of decimal places.

8. Common mistakes with quadratics

Teachers and examiners see the same slip ups every year. Avoid these and you already gain marks.

1. Forgetting to set the equation equal to zero before trying to factorise or use the quadratic formula.
2. Sign errors when substituting into the formula, especially if (b) is negative.
3. Only giving one solution when there should be two.
4. Leaving answers unsimplified, like (\tfrac{10}{4}) instead of (\tfrac{5}{2}).
5. Mixing up turning points and intercepts on quadratic graphs.
6. Not checking for context, for example keeping negative lengths or times in worded questions.

Whenever you solve a quadratic, pause for a second and ask:

• Does this make sense in the question?
• Should there be two answers?
• If it is a graph question, can I sketch a quick picture to check?

9. How to revise quadratic equations and graphs

Here is a simple revision routine you can follow.

9.1 Build the basics

1. Make a quick notes page with the three main forms:
   • Standard form (ax^2 + bx + c)
   • Factorised form ((x + p)(x + q))
   • Completed square form (a(x + p)^2 + q)
2. Write out the quadratic formula correctly three times from memory.
3. Practise spotting difference of two squares.

9.2 Practice schedule

• Day 1: Factorising quadratics where (a = 1). Do 10 mixed questions.
• Day 2: Factorising where (a \ne 1) and using them to solve equations.
• Day 3: Quadratic formula practice with at least 8 questions. Include ones with no real solutions.
• Day 4: Completing the square and finding turning points (Higher).
• Day 5: Sketching quadratic graphs from equations and matching graphs to equations.
• Day 6: Mixed past paper questions that involve quadratics in context.

On each day, mark which questions you got wrong and write a quick note in an error book so you do not repeat the same mistake.

9.3 Use past papers

Quadratics show up often in GCSE papers from all the main boards, including AQA, Edexcel and OCR. Use past paper questions so you get used to the style and wording.

• Search for past paper questions on quadratics and graphs.
• Do the questions under timed conditions where possible.
• Check the mark scheme to see exactly how many marks are available for each step.

10. Key takeaways

• A quadratic has an (x^2) term as its highest power.
• You should be comfortable with factorising, the quadratic formula and, for Higher, completing the square.
• Quadratic graphs are parabolas with a turning point and usually up to two x intercepts.
• Context questions with area, rectangles and height time problems are just quadratics disguised as word problems.
• Regular practice using past paper style questions is the fastest way to turn this topic from a weakness into a strength.

Once you are solid with quadratics, link this topic to others like simultaneous equations, inequalities and coordinate geometry. That way, what you learn here pays off across multiple parts of your GCSE Maths paper.