Quadratic Equations: Roots, Discriminant and the Sum-Product Shortcuts

Why Quadratics Keep Appearing

A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. It shows up whenever a quantity gets multiplied by itself: the area of a rectangle whose sides are related, two consecutive numbers whose product is given, or a ball's height over time. That is why exam setters love it — one short equation hides a whole word problem.

The roots are the values of x that make the expression zero, and a quadratic can have at most two of them. You have three tools: factorisation, the quadratic formula, and the sum and product of roots. The third is the quiet winner, because many questions never ask for the roots at all, only for something built from them. Learn all three and choose the fastest one for each question instead of always solving the long way.

1. What are the roots of x² − 7x + 12 = 0?

2. What is the discriminant of 3x² − 5x + 2 = 0?

3. For what value of k does x² − 8x + k = 0 have equal roots?

4. What are the sum and product of the roots of 2x² − 6x + 4 = 0?

5. If α and β are the roots of x² − 5x + 3 = 0, what is α² + β²?

6. Which quadratic equation has roots 3 and −2?

7. What is the nature of the roots of 2x² − 4x + 3 = 0?

8. The product of two consecutive positive integers is 132. What is the smaller integer?