Forming and solving quadratic equations

Areas, Pythagoras and falling objects often lead to quadratic equations. Form the equation, solve it, then decide which solutions make sense.

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Forming the quadratic

Write the fact as an equation, expand, then rearrange so one side is 0.
A rectangle is x cm wide and (x + 4) cm long with area 45 cm2:
x(x + 4) = 45, so x2 + 4x − 45 = 0.

Solve and reject

x2 + 4x − 45 = (x + 9)(x − 5) = 0, so x = −9 or x = 5.
A width cannot be negative, so reject x = −9. The width is 5 cm.
If it does not factorise, use the quadratic formula.

Pythagoras with x

A right-angled triangle has sides x, x + 7 and hypotenuse x + 8.
x2 + (x + 7)2 = (x + 8)2
x2 + x2 + 14x + 49 = x2 + 16x + 64
x2 − 2x − 15 = 0, so (x − 5)(x + 3) = 0 and x = 5.
Worked example

A rectangle is x cm wide and (x + 3) cm long. Its area is 70 cm2. Find x.

  1. x(x + 3) = 70
  2. x2 + 3x − 70 = 0
  3. (x + 10)(x − 7) = 0
  4. x = −10 (reject) or x = 7

Answer: x = 7

Key idea

Form the equation, rearrange to = 0, solve by factorising or the formula, then reject any answer that is impossible in context.

Key words
Quadratic
An expression or equation where the highest power of x is 2, such as x^{2} − 5x + 6.
Equation
A statement with an equals sign that is true for particular values of the letter, such as 3x + 1 = 10.
Formula
A rule linking two or more variables, such as A = πr^{2}.
Factorise
Write an expression as a product by taking out common factors or putting it into brackets.
Hypotenuse
The longest side of a right-angled triangle, opposite the right angle.

All GCSE and International GCSE Mathematics key terms and definitions →

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  1. A rectangle is x cm wide and (x + 7) cm long. Its area is 60 cm2. Find x.

    Show the answer

    Answer: 5 cm

    x2 + 7x − 60 = 0 gives x = 5 (x = −12 is rejected).

  2. A right-angled triangle has shorter sides x cm and (x + 2) cm and hypotenuse (x + 4) cm. Find x.

    Show the answer

    Answer: 6 cm

    x2 + (x + 2)2 = (x + 4)2 simplifies to x2 − 4x − 12 = 0, so x = 6.

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Practise this lesson free (6 questions)All GCSE and International GCSE Mathematics lessons

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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