What is the quadratic formula?

For ax² + bx + c = 0, the quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. It gives every root of the equation and works for any values of a, b and c provided a is not zero.

How do I know how many roots a quadratic has?

Look at the discriminant b² − 4ac. Positive means two distinct real roots, zero means one repeated real root, and negative means no real roots. Computing the discriminant first is the fastest way to know what to expect.

What does it mean when the discriminant is negative?

The quadratic has no real solutions. The parabola never crosses the x axis. This is a valid mathematical answer rather than a failure. Over the complex numbers there are still two roots, and they are always a conjugate pair of the form p + qi and p − qi.

With a = 0 the equation becomes bx + c = 0, which is linear rather than quadratic. It has the single solution x = −c / b. The quadratic formula divides by a, so it cannot be used in this case, and this calculator switches to the linear solution automatically.

What are Vieta's formulas?

For ax² + bx + c = 0 with roots x1 and x2, Vieta's formulas state that x1 + x2 = −b / a and x1 × x2 = c / a. They are a fast way to check a solver's output without substituting the roots back into the full equation.

How do I find the vertex of a quadratic?

The x coordinate of the vertex is −b / 2a. Substitute that value back into ax² + bx + c to get the y coordinate. The calculator returns both, and the x value is also the axis of symmetry.

Why are my answers different from the back of the book?

The most common cause is a sign error when moving terms across the equals sign, which turns a positive c into a negative one. The second is dropping the negative in front of b. Compare your discriminant with the book's, since that usually identifies which of the two went wrong.

Can I use the quadratic formula if the roots are not whole numbers?

Yes. The formula handles any real coefficients and returns whatever the roots happen to be, including fractions and decimals. If the discriminant is not a perfect square you will see decimals or a square root left in the answer, which is expected.

What is a repeated root?

When the discriminant is exactly zero, both roots are the same number. Graphically the parabola touches the x axis and turns back, so the single solution has a multiplicity of two. For x² − 4x + 4 = 0 the repeated root is x = 2.

When should I factorise instead of using the formula?

When the coefficients are integers and the roots are likely to be whole numbers or simple fractions. Factorising x² − 7x + 10 into (x − 5)(x − 2) is faster than substituting into the formula, and it confirms the answer at the same time. The formula is the fallback when factoring does not work cleanly.

How to Use the Quadratic Equation Solver

This solver handles ax² + bx + c = 0 in every case, including the two that trip people up: a negative discriminant and an a value of zero.

Step-by-Step Instructions

1 Enter a, b and c so the equation reads ax² + bx + c = 0. Put every term on one side of the equals sign before you start.
2 Read the discriminant first. It tells you whether you are getting two roots, one repeated root, or a complex pair.
3 Read both roots, then check them with the sum and product figures against minus b over a and c over a.
4 Look at the vertex tiles for the turning point, which is the axis of symmetry and the minimum or maximum of the curve.
5 Use the How this is worked out panel to copy the substitution line by line into your working.

Understanding the Inputs

a (x squared coefficient)

The coefficient of x squared. It must not be zero for the equation to be quadratic, and this calculator tells you plainly when it is.

How to find it: The number multiplying x². For 3x² + 5x − 2 = 0, a is 3. For x² − 7x + 10 = 0, a is 1 and can be left as is.

Why it matters: The quadratic formula divides by a, so a zero value makes the formula undefined and reduces the problem to a linear one that is solved differently.

Type: number · Default: 1

b (x coefficient)

The coefficient of x. Note that the formula uses negative b, so a positive b is substituted with a minus sign in front.

How to find it: The number multiplying x alone. For 3x² + 5x − 2 = 0, b is 5 and enters the formula as −5.

Why it matters: The sign on b is the most common source of a wrong answer here. Keeping it visible while substituting prevents the error.

Type: number · Default: -3

c (constant)

The constant term, which must be on the same side as everything else. If your equation is written as ax² + bx = −c, flip the sign of c first.

How to find it: The number standing alone. For 3x² + 5x − 2 = 0, c is −2. For x² − 5 = 0, b is 0 and can be left blank or set to zero.

Why it matters: The discriminant and the product of the roots both depend on c directly, so getting its sign right is what makes the whole calculation consistent.

Type: number · Default: 2

Tips & Best Practices

✓ Compute the discriminant before the roots to know what kind of answer to expect.
✓ Verify with the sum and product rather than substituting back into the full equation.
✓ If the coefficients are integers, try factorising first, it is often faster.
✓ For a negative discriminant, remember that no real solutions is the correct and complete answer.

Try the Quadratic Equation Solver

by CalculatorPro Tools · Updated 10-12-2026