National 5 Maths worked question

Solve a quadratic using the quadratic formula

Practise using the quadratic formula to find exact roots, then check your substitution and simplification with a full worked solution.

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Question

Solve 2x² − 3x − 4 = 0 exactly using the quadratic formula.

Show the worked solution

Worked solution

a = 2, b = −3 and c = −4
Read the signed coefficients from ax² + bx + c = 0
x = (−(−3) ± √((−3)² − 4(2)(−4))) ÷ (2(2))
Substitute a, b and c into x = −b ±√(b² − 4ac)2a, keeping the negative signs in brackets.
b² − 4ac = 9 + 32 = 41
The product 4ac is negative, so subtracting it adds 32.
x = 3 +√414 or x = 3 −√414
The discriminant has no square factor greater than 1, and the ± gives both exact roots.

Final answer: x = 3 +√414 or x = 3 −√414

Why this method works

The quadratic formula handles a non-unit coefficient and roots that do not simplify to rational numbers. Keeping b and c signed makes −b positive and makes the discriminant 41; the ± then produces both exact roots.

Check your work

Use the root relationships as a check. The two roots have sum 32 = −ba and product −2 = ca, matching the original quadratic.

Common mistakes

Using b = 3 or c = 4 instead of the signed values −3 and −4
Writing the denominator as 2 instead of 2a = 4
Giving only the plus root or converting √41 to a rounded decimal