AI generatedAlgebraGCSE Higher

Interpreting quadratic roots in context

Reject only roots that do not fit the stated context.

  • Put the quadratic equal to zero before factorising or using the quadratic formula.
  • Keep both square-root branches. Do not divide by an expression containing , as this can remove a root.
  • Give exact answers unless rounding is requested. In a context, reject only roots that are not permitted.

Example 1

A rectangle has width cm and length () cm. Its area is 6 cm². Find its width.

Use area = width × length. Expand, then subtract 6 from both sides.Factorise the quadratic.A product is zero only when at least one factor is zero. Set each factor equal to zero and solve.
Keep this root and solve the other factor.Collect the roots in the answer.A width must be positive. Reject the negative root because it is not a possible length.

Example 2

A rectangle has width cm and length () cm. Its area is 12 cm². Find its width.

Use area = width × length. Expand, then subtract 12 from both sides.Factorise the quadratic.A product is zero only when at least one factor is zero. Set each factor equal to zero and solve.
Keep this root and solve the other factor.Collect the roots in the answer.A width must be positive. Reject the negative root because it is not a possible length.

Your turn

Question 1

A rectangle has width cm and length () cm. Its area is 20 cm². Find its width.

Check answer
Use area = width × length. Expand, then subtract 20 from both sides.Factorise the quadratic.A product is zero only when at least one factor is zero. Set each factor equal to zero and solve.
Keep this root and solve the other factor.Collect the roots in the answer.A width must be positive. Reject the negative root because it is not a possible length.
Question 2

A rectangle has width cm and length () cm. Its area is 30 cm². Find its width.

Check answer
Use area = width × length. Expand, then subtract 30 from both sides.Factorise the quadratic.A product is zero only when at least one factor is zero. Set each factor equal to zero and solve.
Keep this root and solve the other factor.Collect the roots in the answer.A width must be positive. Reject the negative root because it is not a possible length.
Question 3

A rectangle has width cm and length () cm. Its area is 42 cm². Find its width.

Check answer
Use area = width × length. Expand, then subtract 42 from both sides.Factorise the quadratic.A product is zero only when at least one factor is zero. Set each factor equal to zero and solve.
Keep this root and solve the other factor.Collect the roots in the answer.A width must be positive. Reject the negative root because it is not a possible length.
Question 4

A rectangle has width cm and length () cm. Its area is 56 cm². Find its width.

Check answer
Use area = width × length. Expand, then subtract 56 from both sides.Factorise the quadratic.A product is zero only when at least one factor is zero. Set each factor equal to zero and solve.
Keep this root and solve the other factor.Collect the roots in the answer.A width must be positive. Reject the negative root because it is not a possible length.
Question 5

A rectangle has width cm and length () cm. Its area is 72 cm². Find its width.

Check answer
Use area = width × length. Expand, then subtract 72 from both sides.Factorise the quadratic.A product is zero only when at least one factor is zero. Set each factor equal to zero and solve.
Keep this root and solve the other factor.Collect the roots in the answer.A width must be positive. Reject the negative root because it is not a possible length.
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