Quadratic Formula GCSE Questions and Worked Answers

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Start with the equation being solved

What you need to know about the quadratic formula

In x² + 3x − 10 = 0, x is an unknown number and x² means x × x. Try x = 2: 4 + 6 − 10 = 0, so 2 is a solution. Solving means finding all numbers that make the equation true, not just one that works. An equation whose highest power of x is 2 is quadratic. The quadratic formula finds its solutions without trial and error, even when they are not whole numbers.

Equation → coefficients → solutions

Read one equation before using the formula

First write the equation as ax² + bx + c = 0. The letters a and b are the signed numbers multiplying x² and x; c is the number without x. For 2x² − 5x − 4 = 0, a = 2, b = −5 and c = −4. The ± symbol means do one calculation with plus and another with minus.

Standard form2x² − 5x − 4 = 0all terms on one side

Coefficientsa = 2, b = −5, c = −4copy every sign

Formulax = (−b ± √(b²−4ac)) ÷ (2a)the whole top is divided by 2a

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Solve a quadratic that does not factorise neatly

What the problem asks: Find numerical or exact roots of an equation containing x².

How to solve it: Rearrange to zero, identify a, b and c, then substitute into the formula and evaluate both signs.

Solve an equation that needs rearranging

What the problem asks: The x², x and constant terms begin on both sides.

How to solve it: Move every term to one side first. Only then read a, b and c from ax² + bx + c = 0.

Find graph intercepts

What the problem asks: Find where a quadratic graph crosses the x-axis.

How to solve it: At an x-axis crossing y = 0, so set the quadratic expression equal to zero and solve.

Solve a contextual quadratic

What the problem asks: Lengths, areas or another relationship produce an x² equation.

How to solve it: Form the equation, solve both roots, then reject any value that is impossible in the context.

Use the discriminant

What the problem asks: Decide how many real solutions a quadratic has.

How to solve it: Calculate b² − 4ac. Positive gives two real roots, zero gives one repeated root and negative gives no real roots.

A reliable routine

Method for using the quadratic formula

Use this for ax² + bx + c = 0 with a ≠ 0, especially when factorising is not straightforward. Completing the square on this general equation produces the formula. If the number under the square root is negative, there are no real-number solutions.

Rearrange the equation into ax² + bx + c = 0.
Write a, b and c with their signs.
Substitute using brackets: x = (−b ± √(b² − 4ac)) / (2a).
Calculate the square-root expression before dividing by 2a.
Evaluate both the plus and minus cases, then round only at the end if requested.

Check: The whole numerator is divided by 2a. On a calculator, use brackets around both the numerator and denominator.

Fully worked

Quadratic formula GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Find two exact solutions

4 marks

Question

Solve $x^2+3x-1=0$ . Give exact answers.

Here $a=1$ , $b=3$ and $c=-1$ .

x=\frac{-3\pm\sqrt{3^2-4(1)(-1)}}{2(1)}

x=\frac{-3\pm\sqrt{13}}{2}

\boxed{x=\frac{-3+\sqrt{13}}2\text{ or }x=\frac{-3-\sqrt{13}}2}

Example 2

Handle a coefficient greater than one

4 marks

Question

Solve $2x^2+5x-3=0$ .

Here $a=2$ , $b=5$ and $c=-3$ .

x=\frac{-5\pm\sqrt{5^2-4(2)(-3)}}{2(2)}

x=\frac{-5\pm\sqrt{49}}4

x=\frac{-5\pm7}{4}

\boxed{x=\frac12\text{ or }x=-3}

Example 3

Round only at the end

4 marks

Question

Solve $3x^2-4x-2=0$ . Give each answer to 3 significant figures.

Here $a=3$ , $b=-4$ and $c=-2$ .

x=\frac{-(-4)\pm\sqrt{(-4)^2-4(3)(-2)}}{2(3)}

x=\frac{4\pm\sqrt{40}}6

x\approx1.72076\text{ or }x\approx-0.387426

\boxed{x=1.72\text{ or }x=-0.387\quad(3\text{ s.f.})}

Example 4

Rearrange before substituting

4 marks

Question

Solve $5x^2=7x+2$ . Give exact answers.

Move every term to the left.

5x^2-7x-2=0

So $a=5$ , $b=-7$ and $c=-2$ .

x=\frac{7\pm\sqrt{(-7)^2-4(5)(-2)}}{10}

\boxed{x=\frac{7+\sqrt{89}}{10}\text{ or }x=\frac{7-\sqrt{89}}{10}}

Example 5

Choose the valid contextual root

5 marks

Question

A rectangle has width $x$ cm, length $(x+4)$ cm and area $45$ cm². Find its dimensions.

Area is width multiplied by length.

x(x+4)=45

x^2+4x-45=0

Using the formula gives

x=\frac{-4\pm\sqrt{4^2-4(1)(-45)}}2

x=\frac{-4\pm14}{2}=5\text{ or }-9

A length cannot be negative, so $x=5$ .

\boxed{\text{width }5\text{ cm, length }9\text{ cm}}

10 original questions · total 34 marks

Quadratic formula GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 40 minutes · Write a, b and c first, show the full substitution and keep unrounded calculator values until the final line. · answers start collapsed

1

Identify the coefficients

2 marks

For $4x^2-7x+2=0$ , state $a$ , $b$ and $c$ .

Show worked answer

Match the equation with $ax^2+bx+c=0$ .

\boxed{a=4,\quad b=-7,\quad c=2}

2

Substitute with a negative b

2 marks

Write the quadratic-formula substitution for $x^2-6x+1=0$ . You do not need to simplify.

Show worked answer

Here $a=1$ , $b=-6$ and $c=1$ .

\boxed{x=\frac{-(-6)\pm\sqrt{(-6)^2-4(1)(1)}}{2(1)}}

3

Solve exactly

4 marks

Solve $x^2-2x-7=0$ . Give exact answers.

Show worked answer

x=\frac{2\pm\sqrt{(-2)^2-4(1)(-7)}}2

x=\frac{2\pm\sqrt{32}}2

x=\frac{2\pm4\sqrt2}{2}

\boxed{x=1+2\sqrt2\text{ or }x=1-2\sqrt2}

4

Give decimal solutions

4 marks

Solve $2x^2+x-4=0$ to 3 significant figures.

Show worked answer

x=\frac{-1\pm\sqrt{1^2-4(2)(-4)}}4

x=\frac{-1\pm\sqrt{33}}4

\boxed{x=1.19\text{ or }x=-1.69\quad(3\text{ s.f.})}

5

Form an area equation

4 marks

A rectangle has sides $x$ cm and $(x+3)$ cm. Its area is $18$ cm². Find $x$ .

Show worked answer

x(x+3)=18

x^2+3x-18=0

Here $a=1$ , $b=3$ and $c=-18$ .

x=\frac{-3\pm\sqrt{3^2-4(1)(-18)}}2

x=\frac{-3\pm9}{2}

So $x=3$ or $x=-6$ .

Reject the negative length.

\boxed{x=3\text{ cm}}

6

Find no real solutions

3 marks

Show that $3x^2+2x+5=0$ has no real solutions.

Show worked answer

Calculate the discriminant.

b^2-4ac=2^2-4(3)(5)

=4-60=-56

It is negative, so $\boxed{\text{there are no real solutions}}$ .

7

Find a repeated root

3 marks

Solve $x^2-8x+16=0$ using the quadratic formula.

Show worked answer

x=\frac{8\pm\sqrt{(-8)^2-4(1)(16)}}2

x=\frac{8\pm\sqrt0}{2}

\boxed{x=4}

Both formula branches give the same repeated root.

8

Find graph intercepts

3 marks

Find the x-intercepts of $y=x^2+x-6$ .

Show worked answer

At an x-intercept, $y=0$ .

x^2+x-6=0

x=\frac{-1\pm\sqrt{1^2-4(1)(-6)}}2

x=\frac{-1\pm5}{2}

So $x=2$ or $x=-3$ .

\boxed{(2,0)\text{ and }(-3,0)}

9

Solve a consecutive-number problem

5 marks

Two consecutive positive integers have product $72$ . Find them.

Show worked answer

Let the smaller integer be $x$ , so the next is $x+1$ .

x(x+1)=72

x^2+x-72=0

x=\frac{-1\pm\sqrt{1^2-4(1)(-72)}}2

x=\frac{-1\pm17}{2}

So $x=8$ or $x=-9$ .

The positive solution is $x=8$ .

\boxed{8\text{ and }9}

10

Make the discriminant zero

Higher only4 marks

The equation $x^2+kx+9=0$ has one repeated real root. Find the possible values of $k$ .

Show worked answer

A repeated root means the discriminant is zero.

k^2-4(1)(9)=0

k^2-36=0

k^2=36

\boxed{k=6\text{ or }k=-6}

Examiner-style feedback

Common the quadratic formula mistakes

Dropping the sign of b or c

In 2x² − 5x − 3 = 0, b = −5 and c = −3. Write the signs before substituting.

Dividing only the square-root term

The entire numerator −b ± √(b² − 4ac) is divided by 2a.

Giving only one answer

The ± symbol creates two calculations. Unless the roots repeat or context rejects one, report both.

Rounding halfway through

Keep the square root or full calculator value until the final requested accuracy.

30-SECOND RECAP

Choose, carry out, check

Translate the wording before you reach for a rule.

Rearrange to equal zero.
Copy signed a, b and c.
Use brackets in the substitution.
Evaluate both ± cases and round last.

Quick answers

Quadratic formula FAQ

When should I use the quadratic formula?

Use it for a quadratic equation, especially when it does not factorise neatly or decimal answers are requested.

What does ± mean?

It means perform one calculation with plus and another with minus.

What is the discriminant?

It is b² − 4ac, the expression under the square root. Its sign determines the number of real roots.

Do I need to rearrange first?

Yes. The coefficients a, b and c must be read from ax² + bx + c = 0.

Build connected skills

What to revise next

Choose a solving method

Quadratic equations

Start with roots and factorising before applying the formula.

Revise quadratic equations

Form a quadratic

Algebraic fractions

Remove variable denominators and solve the resulting equation.

Revise algebraic fractions

Alternative exact method

Completing the square

Rewrite a quadratic to expose its turning point and roots.

Revise completing the square

Build the equation

Simultaneous equations

Solve linked equations, including linear–quadratic pairs.

Revise simultaneous equations

Exact answers

Surds

Simplify square roots left by the formula.

Revise surds

Content standards

Curriculum and rights review

Reviewed 5 September 2026 against DfE content A18 and current Pearson Edexcel, AQA and OCR GCSE Mathematics specifications. All questions are original.

Official specification references