What is the quadratic formula?

The quadratic formula is x = (-b +/- sqrt(b^2 - 4ac)) / 2a. It finds the values of x where ax^2 + bx + c = 0.

What does the discriminant tell you?

The discriminant (b^2 - 4ac) tells you the number and type of solutions. Positive means two real roots, zero means one repeated root, and negative means two complex roots.

Quadratic Equation Solver - Find Roots Instantly

How the Quadratic Formula Works

Every quadratic equation has the form ax² + bx + c = 0, where a, b, and c are constants and a is not zero. The quadratic formula gives you the exact solutions (roots) for any quadratic equation, no matter how messy the numbers.

x = (-b ± √(b² - 4ac)) / 2a

The key is the discriminant, which is the part under the square root: b² - 4ac. If it is positive, you get two different real roots. If it equals zero, you get one repeated root (the parabola just touches the x-axis). If it is negative, the roots are complex numbers with an imaginary part.

Reading the Graph

The graph above shows your parabola. When a is positive, it opens upward like a cup. When a is negative, it opens downward. The vertex is the highest or lowest point, located at x = -b/2a. The roots (if they are real) are where the curve crosses the x-axis. If the parabola does not cross the x-axis, the roots are complex.

Common Examples

x² - 5x + 6 = 0 factors to (x-2)(x-3), giving roots x=2 and x=3. The equation x² + 1 = 0 has no real roots because the discriminant is -4, giving complex roots x = i and x = -i. The equation x² - 4x + 4 = 0 is a perfect square (x-2)², so x=2 is a repeated root.

What if a equals zero?

If a is zero, it is not a quadratic equation. It becomes a linear equation bx + c = 0, which has one solution: x = -c/b. This calculator will let you know when that happens.

What are complex roots?

Complex roots occur when the discriminant is negative. They involve the imaginary unit i, where i² = -1. Complex roots always come in pairs, like 2 + 3i and 2 - 3i. They are real in the mathematical sense but do not correspond to x-axis crossings on a graph.

How is this used in real life?

Quadratic equations model projectile motion (how high and far a ball travels), optimize areas and volumes, calculate profit maximization in business, and appear in physics, engineering, and computer graphics. Any situation where something accelerates or follows a curved path often involves a quadratic.

Can I use this for homework?

Absolutely. Enter your coefficients and check your work. The discriminant and vertex are shown so you can understand the full picture, not just the answer. Try solving by hand first, then use this to verify.

Quadratic Equation Solver

How the Quadratic Formula Works

Reading the Graph

Common Examples

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