Quadratic Formula
Solve second-degree equation.
What does Quadratic Formula mean?
The quadratic formula solves any second-degree polynomial equation of the form ax² + bx + c = 0. It finds the values of x (called roots or zeros) where the parabola crosses the x-axis. These roots are fundamental in algebra, physics, engineering, and many real-world problems involving projectile motion, optimization, and area calculations.
How to calculate Quadratic Formula
The quadratic formula is x = (−b ± √(b² − 4ac)) / (2a). First compute the discriminant, Δ = b² − 4ac. If Δ > 0, there are two distinct real roots. If Δ = 0, there is exactly one real root (a repeated root). If Δ < 0, there are no real roots (the roots are complex numbers). The vertex of the parabola is at x = −b/(2a), y = a·x² + b·x + c.
FAQ
The discriminant (b² − 4ac) determines the nature of the roots. A positive discriminant means two distinct real roots, zero means one repeated real root, and a negative discriminant means no real roots — the parabola does not cross the x-axis, and the solutions are complex numbers.
If a = 0, the equation becomes bx + c = 0, which is a linear equation, not quadratic. The quadratic formula requires division by 2a, so a = 0 would cause division by zero. Use a simple linear solve (x = −c/b) instead.
The vertex is the highest or lowest point on the parabola. For a > 0, the parabola opens upward and the vertex is the minimum. For a < 0, it opens downward and the vertex is the maximum. The vertex x-coordinate is −b/(2a), and you substitute that back into the equation to find the y-coordinate.
It appears in physics (projectile trajectory, stopping distance), engineering (structural load calculations), finance (break-even analysis), and computer graphics (ray-object intersection). Any time a relationship involves a squared variable, the quadratic formula can find critical values.
When the discriminant is negative, the square root of a negative number is involved. The roots become complex numbers of the form x = (−b ± i√|Δ|) / (2a), where i is the imaginary unit (√−1). This calculator shows "No real roots" in that case.
Related calculators
- Logarithm— Inverse exponent function.
- Ratio— Relative comparison of values.
- Proportion— Equality of two ratios.
- Factorial— Product of descending integers.