The Quadratic Formula

When a quadratic will not factor with clean integers, the formula always works: for ax² + bx + c = 0, x = [ −b ± √(b² − 4ac) ] / (2a). The quantity under the root, D = b² − 4ac, is the discriminant, and it tells you how many real solutions exist before you compute anything: D > 0 gives two distinct real roots, D = 0 gives exactly one (a repeated) root, and D < 0 gives no real roots. That last fact is a favourite GRE Quantitative Comparison hook — if a quantity is defined as "a solution to" an equation whose discriminant is negative, the comparison may be "cannot be determined" or the setup may be impossible. Keep the ± as two separate cases and be careful with signs when b is negative: −b of −6 is +6. The on-screen calculator handles the square root, but if b² − 4ac is a perfect square you rarely need it — that is usually the exam quietly telling you the thing would have factored.

✅ Solved examples

✏️ Practice — try these, take hints as needed

📝 Topic test — 8 questions

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