Quadratic Formula Calculator
x = (-b plus or minus the square root of (b squared - 4ac)) / 2a.
Enter the three coefficients
Write your equation in the standard shape, a x squared plus b x plus c equals 0, then read the three coefficients straight off it. The signs travel with the numbers: in x squared minus 5x plus 6, a is 1, b is negative 5 and c is positive 6.
A missing term just means a coefficient of zero. The equation x squared minus 9 has a of 1, b of 0 and c of negative 9. Typing the zero in keeps the formula honest.
Roots, discriminant and vertex
The result gives the roots first, since that is usually what you came for. Below them sit the discriminant and the vertex, which are the two numbers a teacher tends to ask about next.
The discriminant is the quantity under the square root. Its sign alone tells you how many real roots to expect before you finish the arithmetic, which makes it a useful early check.
The formula and what the discriminant decides
The quadratic formula is x equals negative b, plus or minus the square root of b squared minus 4ac, all divided by 2a. Everything the calculator reports comes from that one expression.
The piece under the root, b squared minus 4ac, is the discriminant. Positive means two separate real roots. Exactly zero means the two roots meet at one point. Negative means there are no real roots, only a matched pair of complex ones.
A worked example
For x squared minus 5x plus 6, the discriminant is 25 minus 24, which is 1. A positive value, so two real roots. The formula gives 5 plus or minus 1, over 2, landing on 3 and 2.
The vertex sits halfway between the roots, at x equals 2.5, where y dips to negative 0.25. That is the lowest point of this upward-opening parabola.
Why a cannot be zero, and other slips
If a is zero, the squared term vanishes and the equation is a straight line, not a parabola. The formula also divides by 2a, so a zero there has nothing to divide into. The calculator stops and says so.
The other frequent slip is dropping a sign when copying b or c. Because both feed into the formula directly, a stray minus flips the roots, so it is worth a second glance at the signs before trusting the answer.
Reading complex roots
When the discriminant is negative, its square root is imaginary, and the roots come back as a real part plus or minus an imaginary part written with an i. The parabola in this case never touches the x-axis.
The two complex roots are always mirror images, sharing the same real part with opposite imaginary signs. That pairing is why a quadratic with real coefficients can never have just one complex root on its own.
Quadratic Formula Calculator FAQ
What is the quadratic formula?
It is x equals negative b, plus or minus the square root of b squared minus 4ac, all over 2a.
It solves any equation in the form a x squared plus b x plus c equals 0.
What does the discriminant tell me?
The discriminant is b squared minus 4ac. A positive value means two real roots, zero means one repeated root, and a negative value means two complex roots.
You can read the number of real roots from its sign before finishing the calculation.
Can a quadratic have no real solutions?
Yes. When the discriminant is negative, both roots are complex and the parabola never crosses the x-axis.
The calculator still solves it and shows the complex pair.
Why can a not be zero?
With a of zero there is no squared term, so the equation is linear rather than quadratic.
The formula also divides by 2a, which would be a division by zero.
What is the vertex of a parabola?
It is the turning point, the lowest spot on an upward curve or the highest on a downward one.
Its x value is negative b over 2a, and the calculator works out the matching y value for you.