National 5 Maths worked question

Solve a quadratic using the quadratic formula

A fixed, reviewed quadratic formula question with a complete worked solution and method explanation.

Standard3 marksCalculator

Question

Use the quadratic formula to solve x² + 4x − 1 = 0. Give your answers in exact form.

Show the worked solution

Worked solution

  1. a = 1, b = 4 and c = −1

    Read the coefficients from ax² + bx + c = 0

  2. x = (−4 ± √(4² − 4(1)(−1))) ÷ (2(1))

    Substitute a, b and c into the quadratic formula.

  3. x = (−4 ± √20) ÷ 2

  4. x = (−4 ± 2√5) ÷ 2 = −2 ± √5

    Simplify √20 to 2√5 and divide both terms by 2.

Final answer: x = −2 + √5 or x = −2 − √5

Why this method works

The signs of b and c must be carried into the quadratic formula. The ± symbol produces two roots, and the exact surd should be simplified before the final answer is stated.

Check your work

The two roots have sum −4 and product −1, matching − and for the original quadratic.

Common mistakes

  • Using c = 1 instead of c = −1
  • Forgetting the denominator 2a.
  • Giving only one of the two roots.