Quadratic Formula Calculator — Roots, Vertex, Factored Form
Type a, b, c. Get the roots (real or complex), discriminant, vertex, axis of symmetry, and the factored form when it's clean.
ax² + bx + c = 0
- Discriminant
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- Vertex
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- Axis of symmetry
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- Factored form
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What is quadratic formula calculator?
The quadratic formula is the closed-form solution to ax² + bx + c = 0. Every quadratic equation has exactly two roots (counted with multiplicity), and the formula gives them in one step from the three coefficients: x = (-b ± √(b² - 4ac)) / (2a). The expression under the square root is the discriminant — its sign tells you whether the roots are two distinct reals, one repeated real, or a pair of complex conjugates.
This calculator returns more than just the roots. The discriminant comes through as a numeric value (handy for sanity-checking your factoring). The vertex (h, k) and the axis of symmetry locate the parabola for graphing — h is always -b/(2a) and the parabola opens up when a > 0, down when a < 0. The factored form is shown when both roots are integer-rational; for irrational or complex roots the factored line is suppressed to keep the output clean.
Privacy is simple: every calculation runs in your browser. The coefficients you type never leave your device, no state is saved between visits, and there are no analytics on input values.
When to use a quadratic formula calculator
- Algebra homework — find the two roots — Type a, b, c. The headline shows both roots (separated by a comma when distinct, or the repeated form when the discriminant is zero). For complex roots, the calculator formats them as <code>a ± bi</code> with the imaginary unit explicit.
- Pre-calc — vertex form for graphing — Need to plot the parabola? The vertex (h, k) and the axis of symmetry x = h appear in the stats list. Use those plus the y-intercept (which is just c) to sketch the curve in three points.
- Factoring check — If the polynomial factors cleanly into integer-rational roots, the factored form appears in the stats list. If it doesn't (irrational or complex roots), the factored form is suppressed — your teacher probably wants you to leave the radical form.
How to use the Quadratic Formula Calculator — Roots, Vertex, Factored Form
- Type the three coefficients — Coefficient a goes with x², b goes with x, c is the constant term. Negative values are allowed (use a minus sign in the input). The equation preview at the top of the widget updates as you type so you can sanity-check your input.
- Read the roots in the headline — Two distinct real roots are shown separated by a comma. A repeated root is shown as <code>x = r (repeated)</code>. Complex roots are formatted as <code>a ± bi</code>.
- Use the stats list for graphing — The discriminant tells you the nature of the roots: positive → two distinct real, zero → one repeated, negative → complex. The vertex (h, k) and axis of symmetry x = h locate the parabola's minimum or maximum and its line of symmetry. The factored form is shown when both roots are integer-rational.
- Copy the summary — Tap the copy button to put a one-line summary of the result on your clipboard — useful when you're moving back and forth between the calculator and your worksheet.
Worked examples
x² - 5x + 6 = 0
Input: a = 1, b = -5, c = 6
Output: x = 3, x = 2 | discriminant = 1, vertex = (2.5, -0.25), factored = (x - 3)(x - 2)Both roots are integers; the factored form is clean.
x² + 1 = 0 (no real roots)
Input: a = 1, b = 0, c = 1
Output: x = ±i | discriminant = -4, vertex = (0, 1), factored = —Discriminant is negative, so the roots are complex conjugates ±i. The factored form is hidden because it would require complex factors.
2x² - 8x + 6 = 0
Input: a = 2, b = -8, c = 6
Output: x = 3, x = 1 | discriminant = 16, factored = 2(x - 3)(x - 1)When the leading coefficient isn't 1, it appears in front of the factored form.
Frequently asked questions
What is the quadratic formula?
x = (-b ± √(b² - 4ac)) / (2a). It gives both roots of a quadratic equation ax² + bx + c = 0 directly from the coefficients. The expression under the square root, b² - 4ac, is the discriminant: positive means two distinct real roots, zero means one repeated real root, negative means two complex conjugate roots.What does the discriminant tell me?
Δ > 0 → two distinct real roots. Δ = 0 → one repeated real root (the parabola is tangent to the x-axis). Δ < 0 → no real roots; the two solutions are complex conjugates of the form a ± bi.How is the vertex calculated?
y = ax² + bx + c sits at (h, k) where h = -b / (2a) and k = c - b²/(4a) (which is the y-value at x = h). When a > 0 the vertex is the minimum; when a < 0 it's the maximum.Why does the factored form sometimes show as '—'?
What is i in the complex-root output?
i is the imaginary unit, defined as i = √(-1). When the discriminant is negative, the square-root step pulls out an i: e.g., for x² + 1 = 0, x = ±√(-1) = ±i. The output is formatted as a + bi and a - bi — the two complex conjugate roots.Can a be 0?
bx + c = 0, which is linear, not quadratic — there's at most one solution, and the quadratic formula's denominator (2a) becomes 0. The calculator displays a hint when you set a to 0; clear it or type a non-zero value to proceed.Are my numbers stored or sent anywhere?
localStorage writes.