TFT

Quadratic Equation Solver

Solve equations in the form: ax² + bx + c = 0

ax² + bx + c = 0

How to Use This Quadratic Equation Solver

1

Enter your coefficients

Input the values for a, b, and c from your equation ax² + bx + c = 0. Make sure a is not zero.

2

Click Solve

The calculator will compute the discriminant and find the roots using the quadratic formula.

3

Review the results

You will see both roots (x₁ and x₂), the discriminant value, the nature of roots, and the vertex of the parabola.

Understanding Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The graph of a quadratic equation is a parabola.

x = (-b ± √(b² - 4ac)) / 2a

The quadratic formula gives you the roots (solutions) of any quadratic equation. The ± symbol means there are typically two solutions: one using addition and one using subtraction.

The Discriminant and Nature of Roots

Discriminant (Δ)Nature of RootsGraph Behavior
Δ > 0Two distinct real rootsParabola crosses x-axis twice
Δ = 0One repeated real rootParabola touches x-axis once
Δ < 0Two complex conjugate rootsParabola does not cross x-axis

The discriminant (Δ = b² - 4ac) tells you what kind of solutions to expect before solving.

Finding the Vertex

The vertex is the highest or lowest point on a parabola. For a quadratic equation, the vertex coordinates (h, k) are:

h = -b / 2a
k = f(h) = a(h)² + b(h) + c

If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, the parabola opens downward and the vertex is a maximum.

Common Quadratic Equation Examples

x² - 5x + 6 = 0

a=1, b=-5, c=6 → Roots: x=2, x=3

2x² + 4x - 6 = 0

a=2, b=4, c=-6 → Roots: x=1, x=-3

x² - 4x + 4 = 0

a=1, b=-4, c=4 → One root: x=2 (repeated)

x² + x + 1 = 0

a=1, b=1, c=1 → Complex roots: -0.5 ± 0.866i

Frequently Asked Questions

What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It solves any quadratic equation ax² + bx + c = 0. The formula comes from completing the square on the general quadratic equation.
What does the discriminant tell me?
The discriminant (b² - 4ac) reveals the nature of the roots. Positive means two real roots, zero means one repeated root, and negative means two complex roots. You can check this before solving to know what to expect.
Can a be zero in a quadratic equation?
No. If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula would also fail because it divides by 2a. Make sure your equation has an x² term.
What is the vertex of a parabola?
The vertex is the turning point of the parabola — its highest or lowest point. The x-coordinate is -b/2a, and you find the y-coordinate by plugging this back into the equation. The vertex tells you the maximum or minimum value of the quadratic function.
When do quadratic equations have complex roots?
Complex roots occur when the discriminant is negative (b² - 4ac &lt; 0). This happens when the parabola does not cross the x-axis. The roots come in conjugate pairs: a + bi and a - bi, where i is the imaginary unit (√-1).

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