Quadratic Formula Calculator
Solve ax² + bx + c = 0
Discriminant (b²−4ac)
1
Root Type
2 Real Roots
Root x&sub1;
3
Root x&sub2;
2
Vertex
(2.5, −0.25)
Axis of Symmetry
x = 2.5
Step-by-Step Solution
How the Quadratic Formula Works
The quadratic formula is one of the most important formulas in algebra. It solves any equation of the form ax² + bx + c = 0, where a, b, and c are known numbers and a is not zero. The formula is: x = (−b ± √(b² − 4ac)) / (2a). It always works, regardless of whether the roots are rational, irrational, or complex.
The key to understanding the quadratic formula is the discriminant, Δ = b² − 4ac. When Δ > 0, the equation has two distinct real roots. When Δ = 0, there is exactly one real root (a repeated root). When Δ < 0, both roots are complex conjugates, involving the imaginary unit i = √(−1). This calculator displays complex roots in a + bi form when the discriminant is negative.
The vertex of the parabola y = ax² + bx + c is located at x = −b/(2a), which is the axis of symmetry. The y-coordinate of the vertex equals c − b²/(4a). If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum. This calculator shows the complete solution process step by step, making it ideal for learning and homework verification.
Formula
x = (−b ± √(b² − 4ac)) / 2a
Where:
- a, b, c = coefficients in the equation ax² + bx + c = 0
- b² − 4ac = discriminant (Δ), determines the nature of roots
- If Δ > 0: two distinct real roots
- If Δ = 0: one repeated real root
- If Δ < 0: two complex conjugate roots
Example Calculation
Scenario: Solve 2x² + 5x − 3 = 0
- Step 1: a = 2, b = 5, c = −3
- Step 2: Discriminant = 5² − 4(2)(−3) = 25 + 24 = 49
- Step 3: x = (−5 ± √49) / (2 × 2) = (−5 ± 7) / 4
- Step 4: x₁ = (−5 + 7) / 4 = 0.5, x₂ = (−5 − 7) / 4 = −3
- Result: x = 0.5 or x = −3