Solve ax² + bx + c = 0 with full step-by-step working, including complex roots.
x = (-b ± √(b²-4ac)) / 2a
The discriminant (b²-4ac) tells you the nature of the roots: positive gives two real roots, zero gives one repeated real root, negative gives two complex roots.
Every quadratic equation ax² + bx + c = 0 is solved using the quadratic formula, and buried inside that formula is a specific expression — b² − 4ac, called the discriminant — that reveals how many real solutions the equation has before the rest of the calculation is even finished. A positive discriminant means two distinct real solutions exist (the parabola crosses the x-axis at two separate points). A discriminant of exactly zero means there's precisely one real solution, a repeated "double root" (the parabola just touches the x-axis at a single point, tangent to it rather than crossing it). A negative discriminant means there are no real solutions at all — only two complex conjugate solutions, because the formula would require taking the square root of a negative number.
This matters beyond pure algebra because the number and nature of a quadratic's roots has direct physical meaning in the real-world situations quadratics model — projectile motion, area optimization, and profit-maximization problems, among others. A negative discriminant in a real-world quadratic model (like when a projectile reaches a certain height) means that scenario genuinely never happens under the given conditions, not that there's a calculation error to find. Checking the discriminant first, before grinding through the full quadratic formula, is a fast way to know in advance whether real solutions exist at all, and how many to expect once the calculation is complete.
The discriminant is b² − 4ac, the expression inside the square root of the quadratic formula. Its sign reveals the type and number of solutions before finishing the calculation: positive means two distinct real solutions, zero means one repeated real solution, and negative means no real solutions (only complex ones).
It means the parabola never crosses or touches the x-axis at all — the entire curve stays either completely above or completely below it. There are no real x-values where the equation equals zero, only complex (non-real) solutions.
Because the parabola is tangent to the x-axis — it touches at exactly one point rather than crossing through two separate points. Mathematically, both roots produced by the quadratic formula turn out to be identical, which is why it's called a 'repeated' or 'double' root rather than two distinct solutions.
Because quadratics model real physical situations (projectile height, profit optimization, area problems), and a negative discriminant in that context means the scenario in question genuinely doesn't occur under the given conditions — not that there's a calculation mistake. Checking the discriminant first quickly reveals whether a real-world solution exists before working through the full formula.