Calculadora da fórmula de Bhaskara

The quadratic formula solves any equation of the form ax² + bx + c = 0 for x: x = (−b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is called the discriminant — it determines how many real solutions exist.

If the discriminant is positive, there are two distinct real roots. If it is exactly zero, there is one repeated real root (the parabola touches the x-axis at a single point). If it is negative, there are no real roots — the solutions are complex numbers, and this calculator reports "no real roots" in that case.

Example: for x² − 3x + 2 = 0 (a = 1, b = −3, c = 2), the discriminant is 9 − 8 = 1, giving two real roots: x = 1 and x = 2. You can check this by factoring: (x − 1)(x − 2) = 0.

The vertex of parabola y = ax² + bx + c sits at x = −b/(2a) — useful for optimization problems even when you only need roots here.

Completing the square and factoring are alternative methods; the quadratic formula always works when a ≠ 0.

Worked example with defaults a=1, b=−3, c=2: discriminant 9−8=1 → roots x=1 and x=2. Check: (x−1)(x−2)=0.

Projectile motion height versus time is quadratic when air resistance is ignored — roots of −16t²+v₀t+h₀=0 (US feet) give time when object hits ground; vertex time is −b/(2a) for max height. Projectile −16t²+v₀t+h₀ roots give ground impact time — vertex at −b/(2a) gives max height in feet when air resistance is ignored.

Complex roots when the discriminant is negative mean the parabola never crosses the x-axis — graph y = x² + 1 to see no real intercepts even though the quadratic formula still returns complex values in symbolic algebra courses and computer algebra systems. Physics problems with no real roots still model valid motion when time stays within allowed bounds.

If a=0 the equation is linear, not quadratic — entering b=−3, c=2 with a=0 returns degenerate behavior; ensure a≠0 for parabola roots (defaults a=1 avoid this). a=0 makes the equation linear — defaults a=1, b=−3, c=2 keep a true parabola with discriminant one and roots one and two.