How to Solve a Quadratic Equation Step by Step

A quadratic equation is any equation that can be written in the standard form ax squared plus bx plus c equals zero, where a is not zero. The letters a, b, and c are the coefficients: a multiplies the squared term, b multiplies the linear term, and c is the constant. Because the highest power of the variable is two, a quadratic can have up to two real solutions, called roots.

Geometrically, plotting y equals ax squared plus bx plus c produces a parabola. The real roots of the equation are exactly the points where that parabola crosses the horizontal axis. A parabola can cross twice, touch once, or never touch, which is why a quadratic can have two, one, or no real solutions.

The quadratic formula solves every quadratic: x equals negative b plus or minus the square root of b squared minus four a c, all divided by two a. You can memorize it and apply it mechanically once the equation is in standard form. The part under the square root, b squared minus four a c, is called the discriminant and it tells you how many real solutions to expect before you finish.

If the discriminant is positive there are two distinct real roots. If it is exactly zero there is one repeated real root, where the parabola just touches the axis. If it is negative there are no real roots, only a pair of complex ones. Checking the discriminant first is a quick way to know what kind of answer you are heading toward.

The quadratic formula always works, but factoring is faster when the numbers are friendly. To factor, look for two numbers that multiply to a times c and add to b, then split the middle term and group. If the equation factors cleanly into two binomials, setting each factor equal to zero gives the roots directly.

Completing the square is the method the quadratic formula is derived from, and it is useful when you need the vertex form of a parabola. You move the constant aside, add the square of half the linear coefficient to both sides, and rewrite the left side as a perfect square. For everyday solving, the formula is usually the quickest reliable path.

The calculator runs entirely in your browser, so the coefficients you type are never uploaded anywhere. It is a good way to confirm work you did by hand and to see the intermediate steps rather than just a final number.

The most frequent error is a sign slip. The formula uses negative b, so if b is already negative the term becomes positive, and the four a c under the root also carries the sign of a and c. Write each substitution out fully before simplifying rather than doing it in your head.

Another trap is forgetting the plus or minus, which quietly drops one of the two roots. Also make sure the equation is in standard form first; if there are terms on both sides, move everything to one side so the other side is genuinely zero before you read off a, b, and c.