Topic guide · Algebra

Quadratic equations: a practical foundation

A quadratic equation has the form ax² + bx + c = 0, where a is not zero. The structure of its roots can reveal a lot before you calculate them.

Three ways to approach a quadratic

Factoring is efficient when the coefficients permit it. Completing the square helps explain the shape of a parabola and leads to the quadratic formula. The formula x = (−b ± √(b² − 4ac)) / 2a works for any quadratic, provided you interpret its roots in the relevant number system.

Worked example: factor and check

Solve x² − 7x + 12 = 0. Two numbers that multiply to 12 and add to −7 are −3 and −4. Thus (x − 3)(x − 4) = 0, giving x = 3 or x = 4. Substitute each value in the original equation to check both answers.

Use the discriminant

The discriminant D = b² − 4ac tells us about real roots. If D is positive there are two distinct real roots; if it is zero there is one repeated real root; if it is negative there are no real roots. For x² − 4x + 4 = 0, D = 16 − 16 = 0 and the repeated root is 2.

Sum and product of roots

If the roots are α and β, then α + β = −b/a and αβ = c/a. These relationships can save time when a problem asks about the roots without requiring you to find each one. They also give a useful check on your solution.

Try it yourself

1. Solve x² − 5x + 6 = 0.
2. How many real roots does x² + 2x + 5 = 0 have?
3. Find the sum and product of the roots of 2x² − 3x − 2 = 0.

Answers: 1. x = 2 or 3. 2. None, because D = 4 − 20 = −16. 3. Sum = 3/2 and product = −1.

Always check additional conditions in the question, especially when the quadratic is part of a larger expression with a denominator, logarithm or square root.