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Enter a, b, and c to find the roots of any quadratic equation along with its discriminant and vertex.
Example
Solve x² − 3x + 2 = 0 (a = 1, b = −3, c = 2):
discriminant = (−3)² − 4·1·2 = 9 − 8 = 1
x = (3 ± √1) / 2 → x₁ = 2, x₂ = 1
vertex = (−b/2a, c − b²/4a) = (1.5, −0.25)
Two distinct real roots: x = 2 and x = 1.
How it works
Computes the discriminant b^2-4ac, then applies x = (-b ± √(discriminant)) / (2a); negative discriminants yield complex conjugate roots. The vertex is (-b/2a, c - b^2/4a).
Good to know
This calculator solves any quadratic equation written in the standard form ax² + bx + c = 0. You type the three coefficients — a (the x² term), b (the x term), and c (the constant) — and it instantly returns the two roots, the discriminant, a plain-language label for the kind of roots you have, and the coordinates of the parabola's vertex. It is built for algebra and precalculus students checking homework, teachers building examples, and anyone who needs a fast, reliable solve without graphing software.
Reach for it whenever you would otherwise plug numbers into the quadratic formula by hand: finding where a parabola crosses the x-axis, factoring a stubborn trinomial, solving a projectile-motion or area problem, or confirming an answer before a test. Because it shows the discriminant separately, it also doubles as a quick way to predict how many real solutions an equation has before you commit to solving it.
To read the output, start with the discriminant: a positive value means two distinct real roots, zero means one repeated root, and a negative value produces a complex conjugate pair written with an i term. The vertex (h, k) tells you the parabola's turning point — if a is positive that point is the minimum, and if a is negative it is the maximum. The two roots are always symmetric about the vertical line x = h, which is a handy sanity check on your answers.
One practical caveat: enter coefficients exactly, including the minus sign, since b = −3 and b = 3 give completely different roots. Fractional or decimal inputs are fine, and results are rounded to six decimal places, so a value shown as a clean integer like 2 may actually be a rounded approximation when your coefficients are messy. If you set a to 0 the equation stops being quadratic and the tool quietly switches to solving the linear case x = −c/b instead.
Frequently asked questions
What does the discriminant tell me?
The discriminant b²−4ac determines root nature: positive means two distinct real roots, zero means one repeated real root, and negative means two complex conjugate roots.
What if I enter a = 0?
With a = 0 the equation is no longer quadratic but linear (bx + c = 0), so the calculator solves it as x = −c/b instead and skips the discriminant and vertex.
Is my data uploaded anywhere?
No — this calculator runs entirely in your browser; nothing is uploaded.
Is it free?
Yes, completely free with no sign-up and no limits.
People also ask
How do you find the roots of a quadratic equation?
Use the quadratic formula x = (−b ± √(b²−4ac)) / (2a) after writing the equation in standard form ax² + bx + c = 0. Substitute your three coefficients, compute the value under the square root first, then evaluate the plus and minus cases to get the two roots.
What is the vertex of a parabola and how is it calculated?
The vertex is the turning point of the parabola, with x-coordinate −b/2a and y-coordinate c − b²/4a. It is the minimum point when a is positive and the maximum point when a is negative.
Can a quadratic equation have only one solution?
Yes. When the discriminant b²−4ac equals exactly zero there is one repeated real root at x = −b/2a, and the parabola just touches the x-axis at its vertex instead of crossing it.
What does it mean when a quadratic has complex roots?
It means the discriminant is negative, so the parabola never crosses the x-axis and there are no real solutions. The roots come as a complex conjugate pair of the form p + qi and p − qi, where p = −b/2a.
Is the quadratic formula the same as factoring?
They solve the same equation and give the same roots, but the quadratic formula always works while factoring only works easily when the roots are rational. The discriminant tells you whether a clean factorization is even possible.
How do you write a quadratic equation in standard form?
Move every term to one side so the equation equals zero and arrange it as ax² + bx + c = 0, with a, b, and c being constants and a not equal to zero. Combine like terms first so each power of x appears only once.
What happens if the leading coefficient a is negative?
The parabola opens downward and the vertex is its highest point, but the roots are found the exact same way with the quadratic formula. A negative a does not change whether the roots are real or complex; only the discriminant determines that.
Do quadratic equations always have two roots?
Counting multiplicity and complex numbers, a quadratic always has exactly two roots. They may be two distinct real numbers, one repeated real number, or a pair of complex conjugates, depending on the sign of the discriminant.
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