The discriminant
For a quadratic equation ax² + bx + c = 0, the discriminant is the quantity b² − 4ac. It tells us the nature of the roots without solving the equation.
Key idea
Discriminant Δ = b² − 4ac.
Δ > 0: two distinct real roots.
Δ = 0: two equal (repeated) real roots.
Δ < 0: no real roots.
Using the discriminant
Problems often give a condition such as “equal roots” or “two distinct roots” and ask for an unknown coefficient. Set up the discriminant, apply the correct sign, and solve.
Worked example
Find k so that x² + kx + 9 = 0 has equal roots. Here a = 1, b = k, c = 9, so Δ = k² − 4(1)(9) = k² − 36. For equal roots Δ = 0: k² = 36, giving k = 6 or k = −6.
Remember
- Write the equation as ax² + bx + c = 0 first.
- Δ > 0 two distinct, Δ = 0 equal, Δ < 0 none.
- “Real roots” (at least one) means Δ ≥ 0.