Solve Quadratic Equation
The quadratic formula calculator solves any equation of the form ax² + bx + c = 0. Enter your coefficients below to get roots, discriminant, vertex, step-by-step work, and a graph of the parabola.
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The Quadratic Formula
The quadratic formula solves any quadratic equation ax² + bx + c = 0, where a ≠ 0:
What the parts mean
- a, b, c — the coefficients of x², x, and the constant term
- ± — produces two solutions: one with +, one with −
- b² − 4ac — the discriminant (Δ), which determines the type of roots
Discriminant (Δ = b² − 4ac)
- Δ > 0: Two distinct real roots — the parabola crosses the x-axis at two points
- Δ = 0: One repeated real root — the parabola just touches the x-axis (tangent)
- Δ < 0: Two complex conjugate roots — the parabola does not cross the x-axis
Vertex of the Parabola
The vertex is the highest or lowest point on the parabola. Its x-coordinate is x = −b/(2a), which is also the axis of symmetry. Substitute back into the equation to get the y-coordinate. When a > 0, the vertex is a minimum; when a < 0, it is a maximum.
How the Formula Is Derived (Completing the Square)
- Start with ax² + bx + c = 0
- Divide everything by a: x² + (b/a)x + c/a = 0
- Move the constant: x² + (b/a)x = −c/a
- Add (b/(2a))² to both sides: (x + b/(2a))² = (b² − 4ac) / (4a²)
- Take the square root: x + b/(2a) = ±√(b² − 4ac) / (2a)
- Solve for x: x = (−b ± √(b² − 4ac)) / (2a)
Common Mistakes
- Forgetting that a ≠ 0. If a = 0, the equation is linear (bx + c = 0), not quadratic.
- Sign errors with b. In the formula, the first term is −b, not b. If b is already negative, then −b is positive.
- Dividing only part of the numerator by 2a. The entire expression (−b ± √Δ) is divided by 2a, not just the square root.
- Confusing the discriminant with the roots. The discriminant tells you the type of roots. You still need to finish the formula to get the actual values.
- Rounding too early. Keep the full square root value until the final step to avoid rounding errors.
Frequently Asked Questions
What is the quadratic formula?
The quadratic formula is x = (−b ± √(b² − 4ac)) / (2a). It solves any equation of the form ax² + bx + c = 0, where a ≠ 0. The ± means there are generally two solutions: one using + and one using −.
What is the discriminant and what does it tell you?
The discriminant is Δ = b² − 4ac, the expression under the square root in the quadratic formula. If Δ > 0, the equation has two distinct real roots. If Δ = 0, there is exactly one repeated real root. If Δ < 0, the roots are complex conjugates (involving imaginary numbers).
Can a quadratic equation have no real solutions?
Yes. When the discriminant b² − 4ac is negative, the equation has no real solutions. Instead, it has two complex conjugate roots. For example, x² + 1 = 0 has roots x = i and x = −i.
What happens when a = 0?
When a = 0, the equation becomes bx + c = 0, which is linear, not quadratic. The quadratic formula does not apply because it requires division by 2a. A linear equation has exactly one solution: x = −c/b (when b ≠ 0).
How do I find the vertex of a parabola?
The vertex is at x = −b/(2a). Substitute this x-value back into ax² + bx + c to get the y-coordinate. The vertex is the minimum point when a > 0 (parabola opens upward) and the maximum point when a < 0 (parabola opens downward).
What is the axis of symmetry?
The axis of symmetry is the vertical line x = −b/(2a) that passes through the vertex. Every parabola is symmetric about this line, meaning the two roots are equidistant from it.
How is the quadratic formula derived?
The quadratic formula 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 to form a perfect square, then take the square root and solve for x.
Does this calculator store my data?
No. All calculations run entirely in your browser using JavaScript. No data is sent to any server, and nothing is stored.
Privacy & Limitations
- All calculations run entirely in your browser -- nothing is sent to any server.
- Results are computed using standard formulas and should be verified for critical applications.
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Quadratic Formula Calculator FAQ
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a). It solves any equation of the form ax² + bx + c = 0, where a ≠ 0. The ± means there are generally two solutions: one using + and one using -.
What is the discriminant and what does it tell you?
The discriminant is Δ = b² - 4ac, the expression under the square root in the quadratic formula. If Δ > 0, the equation has two distinct real roots. If Δ = 0, there is exactly one repeated real root. If Δ < 0, the roots are complex conjugates (involving imaginary numbers).
Can a quadratic equation have no real solutions?
Yes. When the discriminant b² - 4ac is negative, the equation has no real solutions. Instead, it has two complex conjugate roots. For example, x² + 1 = 0 has roots x = i and x = -i.
What happens when a = 0?
When a = 0, the equation becomes bx + c = 0, which is linear, not quadratic. The quadratic formula does not apply because it requires division by 2a. A linear equation has exactly one solution: x = -c/b (when b ≠ 0).
How do I find the vertex of a parabola?
The vertex is at x = -b/(2a). Substitute this x-value back into ax² + bx + c to get the y-coordinate. The vertex is the minimum point when a > 0 (parabola opens upward) and the maximum point when a < 0 (parabola opens downward).
What is the axis of symmetry?
The axis of symmetry is the vertical line x = -b/(2a) that passes through the vertex. Every parabola is symmetric about this line, meaning the two roots are equidistant from it.
How is the quadratic formula derived?
The quadratic formula 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 to form a perfect square, then take the square root and solve for x.
Does this calculator store my data?
No. All calculations run entirely in your browser using JavaScript. No data is sent to any server, and nothing is stored.