Solve any ax² + bx + c = 0.
Enter the three coefficients to find the roots, the discriminant, the vertex and the axis of symmetry, updated live as you type.
Your equation
What this means
The parabola around the vertex
| x | y = ax² + bx + c | Position |
|---|
How the quadratic formula works
The Quadratic Formula
The quadratic formula x = (-b ± √(b²-4ac)) / 2a solves any equation of the form ax² + bx + c = 0. It gives up to two solutions (roots) where the parabola crosses the x-axis. This is one of the most important formulas in algebra.
Understanding the Discriminant
The discriminant (b² - 4ac) determines the nature of the roots. If positive, there are two distinct real roots. If zero, there is exactly one real root (a repeated root). If negative, there are two complex conjugate roots and the parabola does not cross the x-axis.
Vertex and Axis of Symmetry
The vertex of the parabola is at x = -b/(2a), y = f(-b/(2a)). The axis of symmetry is the vertical line x = -b/(2a). 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.
Applications of Quadratic Equations
Quadratic equations model projectile motion (height over time), area optimization, revenue/profit functions, braking distance, and many physics problems. The parabolic shape appears in satellite dishes, headlight reflectors, and suspension bridge cables.
Common questions
What happens if a = 0?
If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula cannot be applied. The single solution is x = -c/b.
What are complex roots?
When the discriminant is negative, the roots involve the imaginary unit i (√(-1)). For example, x² + 1 = 0 has roots x = i and x = -i. Complex roots always come in conjugate pairs (a+bi and a-bi).
Can I use this for equations not in standard form?
Yes, first rearrange your equation into ax² + bx + c = 0 form by moving all terms to one side. Then identify the coefficients a, b, and c to enter into the calculator.
How is the quadratic formula derived?
It is derived by completing the square on ax² + bx + c = 0. Divide by a, move c/a to the right, add (b/2a)² to both sides, factor the left as a perfect square, then take the square root and solve for x.
Results are computed with standard floating point arithmetic and rounded for display. For exact symbolic roots, use a computer algebra system.