Quadratic Equation Solver

Q: What is the quadratic formula?

The quadratic formula is x = (−b ± √(b²−4ac)) / (2a). It gives the two roots of any quadratic equation ax² + bx + c = 0 where a ≠ 0. The ± sign means there are usually two solutions: one using + and one using −.

Q: What is the discriminant and what does it tell you?

The discriminant is the expression b² − 4ac inside the square root of the quadratic formula. If discriminant > 0: two distinct real roots. If discriminant = 0: one repeated real root (a perfect square trinomial). If discriminant < 0: two complex (imaginary) conjugate roots, meaning the parabola does not cross the x-axis.

Q: How do I solve a quadratic equation by factoring?

For simple cases, try to find two numbers that multiply to ac and add to b, then factor. For example, x² + 5x + 6 = 0: find numbers that multiply to 6 and add to 5 - those are 2 and 3. So (x+2)(x+3) = 0, giving roots x = −2 and x = −3. When factoring is not obvious, use the quadratic formula instead.

Solve ax² + bx + c = 0 - find real and complex roots instantly.

x² What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree 2, written in the standard form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. The term "quadratic" comes from the Latin word quadratus meaning square, because the highest power of the variable is 2 (squared).

Quadratic equations arise everywhere in mathematics, physics, and engineering. The trajectory of a thrown ball follows a parabolic path described by a quadratic equation. The area of a rectangle with a fixed perimeter is a quadratic function of its dimensions. In electronics, the resonant frequency of an LC circuit involves solving a quadratic. In finance, calculating the internal rate of return can require solving a quadratic. Even the perspective projection in 3D graphics uses quadratic equations.

The graph of y = ax² + bx + c is always a parabola. When a > 0, it opens upward (U-shape); when a < 0, it opens downward (∩-shape). The roots of the equation are the x-coordinates where the parabola crosses the x-axis. The vertex is the highest or lowest point of the parabola, and the axis of symmetry passes through it vertically.

There are four methods to solve quadratic equations: factoring (fast for simple cases), completing the square (foundational technique), the quadratic formula (always works), and graphing (visual but imprecise). The quadratic formula is derived by completing the square on the general form and gives exact solutions - including complex roots - for any quadratic equation.

📐 Quadratic Formula

x = (−b ± √(b² − 4ac)) / (2a)

a = coefficient of x² (must not be 0)

b = coefficient of x

c = constant term

b² − 4ac = discriminant (Δ) - determines nature of roots

The discriminant Δ = b² − 4ac determines how many real solutions exist:

Δ > 0 → Two distinct real roots (parabola crosses x-axis twice)

Δ = 0 → One repeated real root (parabola just touches x-axis)

Δ < 0 → Two complex conjugate roots (parabola does not cross x-axis)

The vertex of the parabola is at (h, k) where h = −b/(2a) and k = c − b²/(4a). The sum of the roots is −b/a and the product of the roots is c/a (Vieta's formulas).

📖 How to Use This Calculator

Steps to Solve a Quadratic Equation

Write your equation in standard form ax² + bx + c = 0. Move all terms to one side so the right side is 0.

Identify a, b, and c. For 2x² − 3x + 1 = 0, a = 2, b = −3, c = 1. Pay attention to signs - a negative b means you enter a negative number.

Enter the values in the a, b, and c fields and click Solve.

Read the roots (x₁ and x₂), discriminant, and vertex. The step-by-step solution shows exactly how the formula was applied.

💡 Example Calculations

Example 1 - Two real roots: x² − 5x + 6 = 0

a = 1, b = −5, c = 6. Discriminant = (−5)² − 4(1)(6) = 25 − 24 = 1 (positive → 2 real roots)

x = (5 ± √1) / 2 = (5 ± 1) / 2

x₁ = (5 + 1) / 2 = 3 | x₂ = (5 − 1) / 2 = 2

Roots: x = 3 and x = 2. Verify: (x−3)(x−2) = x²−5x+6 ✓

Example 2 - One repeated root: x² − 6x + 9 = 0

a = 1, b = −6, c = 9. Discriminant = (−6)² − 4(1)(9) = 36 − 36 = 0 (one repeated root)

x = (6 ± 0) / 2 = 6/2 = 3

Root: x = 3 (repeated). This is a perfect square: (x−3)² = 0. Vertex touches x-axis at (3, 0).

Example 3 - Complex roots: x² + x + 1 = 0

a = 1, b = 1, c = 1. Discriminant = 1² − 4(1)(1) = 1 − 4 = −3 (negative → complex roots)

x = (−1 ± √(−3)) / 2 = (−1 ± i√3) / 2

Roots: x = −0.5 + 0.866i and x = −0.5 − 0.866i (complex conjugates)

❓ Frequently Asked Questions

What is the quadratic formula?+

The quadratic formula is x = (−b ± √(b²−4ac)) / (2a). It gives the two roots of any quadratic equation ax² + bx + c = 0 where a ≠ 0. The ± sign means there are usually two solutions: one using + and one using −. It always works - unlike factoring, which only works easily for simple equations.

What is the discriminant and what does it tell you?+

The discriminant is b² − 4ac inside the square root of the quadratic formula. If discriminant > 0: two distinct real roots - the parabola crosses the x-axis twice. If discriminant = 0: one repeated root - the parabola just touches the x-axis at its vertex. If discriminant < 0: two complex conjugate roots - the parabola doesn't cross the x-axis at all.

How do I solve a quadratic equation by factoring?+

For simple cases, find two numbers that multiply to ac and add to b, then factor. For x² + 5x + 6 = 0: find numbers multiplying to 6 and adding to 5 - those are 2 and 3. Factor: (x+2)(x+3) = 0, giving roots x = −2 and x = −3. When factoring isn't obvious, use the quadratic formula instead - it always works.

Can a quadratic equation have no real solutions?+

Yes. When the discriminant (b²−4ac) is negative, the equation has no real solutions - it has two complex conjugate roots of the form p ± qi where i = √(−1). Graphically, the parabola y = ax²+bx+c does not intersect the x-axis. Example: x² + 1 = 0 has roots ±i (no real solutions).

What is vertex form of a quadratic?+

Vertex form is y = a(x−h)² + k, where (h, k) is the vertex of the parabola. To convert from standard form: h = −b/(2a) and k = c − b²/(4a). The axis of symmetry is the vertical line x = h. Vertex form is useful for graphing: it directly shows the turning point and direction of opening (up if a > 0, down if a < 0).

📌 Quick Tips

💡If the discriminant (b²−4ac) is positive, there are 2 real roots. If zero, there is exactly 1 real root (a repeated root). If negative, the roots are complex (imaginary).

💡Vieta's formulas: the sum of the roots = −b/a, and the product of the roots = c/a. Use these to quickly verify your solutions.

💡A quadratic ax²+bx+c can be factored as a(x−r₁)(x−r₂) where r₁ and r₂ are the roots - useful for factoring by inspection.

🔗 Related Calculators

📌 Quick Tips

💡If the discriminant (b²−4ac) is positive, there are 2 real roots. If zero, there is exactly 1 real root (a repeated root). If negative, the roots are complex (imaginary).

💡Vieta's formulas: the sum of the roots = −b/a, and the product of the roots = c/a. Use these to quickly verify your solutions.

💡A quadratic ax²+bx+c can be factored as a(x−r₁)(x−r₂) where r₁ and r₂ are the roots - useful for factoring by inspection.