Quadratic Functions & Equations
Factoring, the quadratic formula, and the discriminant.
The idea
A quadratic equation has the standard form:
When it factors nicely, set each factor equal to zero. When it doesn't, the quadratic formula always works:
The expression under the square root, b² − 4ac, is called the discriminant. It tells you how many real solutions the equation has: positive → two real solutions, zero → one real solution, negative → no real solutions.
Practice problems
Solve x² − 5x + 6 = 0 by factoring.
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(x − 2)(x − 3) = 0, so x = 2 or x = 3.
Solve 2x² + 3x − 2 = 0 using the quadratic formula.
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a = 2, b = 3, c = −2. Discriminant = 3² − 4(2)(−2) = 9 + 16 = 25. x = (−3 ± 5) / 4, so x = 1/2 or x = −2.
A ball's height is modeled by h(t) = −16t² + 64t + 5 (feet, t in seconds). When does it hit the ground? Round to the nearest hundredth of a second.
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Set h(t) = 0: a = −16, b = 64, c = 5. Discriminant = 64² − 4(−16)(5) = 4096 + 320 = 4416, √4416 ≈ 66.45. t = (−64 ± 66.45) / (−32). The positive root gives t ≈ 4.08 seconds (the negative root is rejected — time can't be negative).
Use the discriminant to determine the number of real solutions of x² + 4x + 9 = 0.
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Discriminant = 4² − 4(1)(9) = 16 − 36 = −20, which is negative. There are no real solutions.