📈 Quadratic Solver
Solve ax² + bx + c = 0 with full step-by-step solution.
Roots
How to Use This Calculator
Enter the three coefficients a, b, and c for your quadratic equation ax² + bx + c = 0. The equation preview updates in real time. Click Solve to get the roots, discriminant, vertex, axis of symmetry, and the direction the parabola opens.
Identify a, b, and c from your equation. For 2x² - 7x + 3 = 0, a = 2, b = -7, c = 3. Make sure the equation is in the form ax² + bx + c = 0 before reading off the values.
Enter a in the first field. Note that a cannot be zero — if it were, the equation would be linear, not quadratic.
Enter b and c. These can be positive, negative, or zero.
Click Solve and read the results. The discriminant tells you before you even look at the roots whether to expect two real roots, one repeated root, or two complex roots.
Quadratic Formula
The discriminant Δ = b² - 4ac tells you everything about the nature of the roots before you calculate them. Δ > 0 means two distinct real roots. Δ = 0 means one repeated root (the parabola just touches the x-axis). Δ < 0 means two complex (imaginary) roots and the parabola does not cross the x-axis at all.
Worked Examples
Where This Comes Up in Real Life
Quadratic equations describe the path of any projectile under constant gravity (parabolic trajectory). If a ball is thrown upward with initial height h₀ = 1.5 m and initial vertical velocity v₀ = 12 m/s, the height at time t is h = -4.905t² + 12t + 1.5. Setting h = 0 gives a quadratic to solve: when does the ball hit the ground? The positive root gives the answer.
Engineers use quadratics when optimising. If a company's profit P = -2x² + 80x - 600 (where x is units produced in hundreds), the vertex gives the production level that maximises profit. Vertex x = -80 / (2 × -2) = 20 (hundreds of units), and the maximum profit is P = -2(400) + 80(20) - 600 = £200. Any calculus-based optimisation problem at its simplest level reduces to solving a quadratic.
Frequently Asked Questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) / (2a). It solves any quadratic equation ax² + bx + c = 0. The ± gives two roots (x₁ and x₂).
What is the discriminant?
The discriminant is Δ = b² − 4ac. If Δ > 0: two distinct real roots. If Δ = 0: one repeated real root. If Δ < 0: two complex (imaginary) roots.
What is the vertex of a parabola?
The vertex is the minimum (a>0) or maximum (a<0) point. Vertex x = −b/(2a), Vertex y = c − b²/(4a) = f(−b/2a).
What are complex roots?
When the discriminant is negative, the roots are complex numbers: x = −b/(2a) ± i√(|Δ|)/(2a), where i = √(−1).
Can I complete the square instead?
Yes — completing the square gives x = −b/(2a) ± √(b²−4ac)/(2a), which is the same as the quadratic formula. It also directly gives the vertex form: a(x−h)²+k.