Quadratic Equations

Equations of the form ax² + bx + c = 0: solve by factorising or the quadratic formula, and use the discriminant to tell what kind of roots there are.

Key ideas

• A quadratic equation can be written as ax² + bx + c = 0 with a ≠ 0. It has at most two roots.
• Factorising: split bx into two terms whose coefficients multiply to ac and add to b, then factorise in pairs.
• If (x − p)(x − q) = 0, then x = p or x = q.
• The quadratic formula works for every quadratic equation.
• The discriminant D = b² − 4ac tells the nature of the roots without solving.
• In word problems, check every root makes sense: a length, speed or age cannot be negative.

Remember

Watch out

• Dividing both sides by x and losing the root x = 0.
• Forgetting the ± and finding only one root.
• Using +b instead of −b in the formula when b is negative.
• Keeping a negative root for a length or an age.

Worked example