Solve ax²+bx+c=0 using three methods: quadratic formula, completing the square, and factoring. Handles real and complex roots. Shows a parabola canvas with roots and vertex marked, vertex form, discriminant analysis, and axis of symmetry.
Type the values of a (x² coefficient, must be non-zero), b (x coefficient), and c (constant). Decimals and negatives are supported. You can also type the full equation in standard form (e.g. "2x²-5x+3=0") and the coefficients are extracted automatically.
Three tabs: Quadratic Formula (always works), Completing the Square (shows the vertex form derivation), and Factoring (when integer factors exist). All three are computed — click any tab to see the full working for that method.
The canvas draws the parabola y=ax²+bx+c with roots (x-intercepts), vertex, and axis of symmetry marked. Displayed values: discriminant (b²−4ac), vertex (h, k), vertex form a(x−h)²+k, x-intercepts, y-intercept, axis of symmetry x=−b/(2a), and domain/range.
For ax² + bx + c = 0 (a≠0), the solutions are: x = (−b ± √(b²−4ac)) / (2a). The ± gives two solutions. The expression b²−4ac is the discriminant (Δ). If Δ > 0: two distinct real roots. If Δ = 0: one repeated real root x = −b/(2a). If Δ < 0: two complex conjugate roots x = (−b ± i√(|Δ|)) / (2a). The formula works for any quadratic regardless of whether it factors over integers, making it the most universal solution method.
Completing the square transforms ax²+bx+c=0 into vertex form a(x−h)²+k=0. Steps: (1) Divide by a: x²+(b/a)x = −c/a. (2) Add (b/2a)² to both sides: x²+(b/a)x+(b/2a)² = (b/2a)² − c/a. (3) Factor left as perfect square: (x+b/2a)² = (b²−4ac)/(4a²). (4) Solve: x = −b/(2a) ± √(b²−4ac)/(2a). This derivation shows WHY the quadratic formula works. The vertex of the parabola is at (h,k) = (−b/2a, c−b²/4a).
A quadratic ax²+bx+c factors into a(x−r₁)(x−r₂) where r₁ and r₂ are the roots. For monic quadratics (a=1): find two numbers that multiply to c and add to b. Example: x²−5x+6: find m×n=6 and m+n=−5 → m=−2, n=−3 → (x−2)(x−3). For a≠1: use the AC method — find two numbers multiplying to ac and adding to b, then factor by grouping. Example: 2x²+7x+3: ac=6, m×n=6 and m+n=7 → m=1, n=6 → 2x²+x+6x+3 = x(2x+1)+3(2x+1) = (2x+1)(x+3).
The discriminant Δ = b²−4ac determines the nature of roots without solving the equation. Δ > 0: two distinct real roots — the parabola crosses the x-axis at two points. Δ = 0: one repeated real root (tangent to x-axis) — the parabola touches the x-axis at exactly one point (vertex on axis). Δ < 0: two complex conjugate roots — the parabola does not cross the x-axis (opens above or below it entirely). The discriminant is also used to determine if a conic section is an ellipse (Δ<0), parabola (Δ=0), or hyperbola (Δ>0) in the general second-degree equation.
The vertex is the highest or lowest point of the parabola y = ax²+bx+c. Its x-coordinate is h = −b/(2a) (also the axis of symmetry). The y-coordinate is k = c − b²/(4a) = f(h). For a>0, the parabola opens upward and the vertex is a minimum. For a<0, it opens downward and the vertex is a maximum. Vertex form: y = a(x−h)²+k clearly shows the vertex and is useful for graphing. Applications: maximizing profit (a<0), minimizing cost (a>0), projectile height as a function of time (a = −g/2 where g is gravity).
Complex roots occur when the discriminant Δ = b²−4ac < 0. They come in conjugate pairs: x = p ± qi where p = −b/(2a) and q = √(|Δ|)/(2a). For example, x²+1=0 → x = ±i. Complex roots mean the parabola never touches the x-axis. In engineering and physics, complex roots of characteristic equations indicate oscillatory (sinusoidal) behaviour — the imaginary part gives the frequency. In control systems, poles with negative real parts (stable) vs positive real parts (unstable) determine system stability.
