Solve ax² + bx + c = 0

What the Quadratic Formula Calculator Does

This calculator solves any quadratic equation written in the standard form ax^2 + bx + c = 0. You enter the three coefficients a, b, and c, and it returns the values of x (called the roots or solutions) that make the equation true.

It is useful for algebra students checking homework, anyone factoring expressions that don't factor cleanly, and people working with projectile motion, area problems, or revenue and break-even calculations in business. The only requirement is that a is not zero. If a = 0, the equation is linear, not quadratic, and the formula does not apply.

How It Works: The Quadratic Formula and Discriminant

The calculator uses the quadratic formula, which gives both roots in one expression:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

The part under the square root, b^2 - 4ac, is called the discriminant. Its sign tells you how many real roots exist before you finish the calculation:

• Discriminant > 0: two distinct real roots (the parabola crosses the x-axis twice).
• Discriminant = 0: one repeated real root (the parabola touches the x-axis at its vertex).
• Discriminant < 0: no real roots; the two solutions are complex conjugates involving the imaginary unit i.

Worked Example with Real Numbers

Solve 2x^2 - 7x + 3 = 0. Here a = 2, b = -7, and c = 3.

First find the discriminant: b^2 - 4ac = (-7)^2 - 4(2)(3) = 49 - 24 = 25. Since 25 is positive, expect two real roots. The square root of 25 is 5.

Now apply the formula: x = (-(-7) ± 5) / (2 × 2) = (7 ± 5) / 4. That gives x = (7 + 5)/4 = 12/4 = 3, and x = (7 - 5)/4 = 2/4 = 0.5. The two roots are x = 3 and x = 0.5. You can verify: 2(3)^2 - 7(3) + 3 = 18 - 21 + 3 = 0, which checks out.

Tips and Common Mistakes

Most errors come from signs and the order of operations rather than the formula itself. Watch these points:

• Use -b, not b. When b is negative, -b becomes positive, as in the example above where -(-7) = 7.
• Square b before subtracting. Because (-7)^2 = 49 is always positive, b^2 is positive even when b is negative.
• Divide the entire numerator by 2a, not just one term. Keep the ± expression grouped.
• Confirm the equation equals zero first. If it reads ax^2 + bx + c = d, move d across so the right side is 0 before reading off a, b, and c.
• Don't forget a hidden coefficient: x^2 - 5x + 6 = 0 has a = 1, not a missing value.

Factors That Affect the Result

The roots depend entirely on the three coefficients, and small changes can shift the outcome between two, one, and zero real solutions. The discriminant is the single quantity that controls this, so a change that pushes b^2 - 4ac across zero changes the nature of the answer.

When the discriminant is a perfect square (like 25), the roots are clean fractions or whole numbers. When it is not, the roots are irrational, and the calculator returns rounded decimals. For a negative discriminant, the calculator reports complex roots in the form p ± qi. If you only need real solutions, a negative discriminant means there are none.