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MathematicsGCSE & IGCSE

What is quadratic formula?

The quadratic formula solves any equation of the form ax² + bx + c = 0, including those that will not factorise.

Quadratic formula explained

Identify aa, bb and cc carefully with their signs, substitute, and simplify. The ±\pm gives the two roots.
The expression under the root, b2−4acb^{2} - 4ac, is the discriminant and tells you how many real roots exist before you solve anything.

Key formula

x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}

Worked example

For 2x2+3x−5=02x^{2} + 3x - 5 = 0: x=−3±9+404=−3±74x = \frac{-3 \pm \sqrt{9 + 40}}{4} = \frac{-3 \pm 7}{4}, so x=1x = 1 or x=−2.5x = -2.5.

Examiner tip

Bracket the negative b when substituting. Writing −(−3)-(-3) rather than 33 prevents the most common sign slip in the whole paper.

Quadratic formula: common questions

When should I use the quadratic formula instead of factorising?
Use it when the expression does not factorise neatly, when the question asks for surd or decimal answers, or when a and b are awkward numbers.
Is the quadratic formula given in the exam?
It is on the GCSE formula sheet for most boards, but you should still know it by heart to save time.

Related terms

Discriminant

The discriminant, b² − 4ac, tells you how many real roots a quadratic equation has without solving it.

Completing the square

Completing the square rewrites a quadratic in the form a(x + p)² + q, revealing the turning point and allowing exact solutions.

Factorising

Factorising rewrites an expression as a product of brackets or factors — the reverse of expanding.

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