Find roots of ax² + bx + c = 0 using the discriminant and quadratic formula.
Quadratic equations appear whenever area, motion, revenue, or optimization models include squared terms. Standard form ax² + bx + c = 0 collects the squared term, linear term, and constant so you can apply systematic solution methods.
The discriminant b² - 4ac tells you how many real roots exist before you finish arithmetic. Positive discriminant means two real roots, zero means one repeated real root, and negative means complex conjugate roots that still satisfy the equation in complex numbers.
The quadratic formula x = (-b ± sqrt(b² - 4ac)) / (2a) works whenever a is not zero. For mental checks, factor when easy (x² - 5x + 6 = (x-2)(x-3)). When factoring is messy, the formula is reliable.
In applied problems, interpret roots as break-even points, intersection times, or dimensions that satisfy constraints. Discard physically impossible roots (negative lengths) even if they are mathematically valid.
Move all terms to one side so ax² + bx + c = 0.
Read coefficients including signs.
Evaluate b² - 4ac to preview root types.
Calculate both ± branches.
Reduce square roots when possible.
Substitute roots back into the original equation.
x = (-b ± sqrt(b² - 4ac)) / (2a)
Derived by completing the square on ax² + bx + c = 0. The discriminant under the square root determines whether roots are real or complex.
Discriminant = 25 - 24 = 1; x = (5 ± 1)/2
Roots: x = 3 and x = 2.
Discriminant = 36 - 36 = 0
Double root x = 3.
Discriminant = 1 - 4 = -3
Complex roots involving sqrt(-3).