Quadratic Formula Calculator - Solve ax² + bx + c = 0

A quadratic formula calculator solves ax² + bx + c = 0 for x when a is not zero. Completing the square once and for all produces

The expression under the radical is the discriminant Δ = b² − 4ac. Its sign decides the geometry of y = ax² + bx + c: two x-intercepts, a vertex sitting on the axis, or a pair of complex conjugate roots with no real crossing. Factoring is faster when the numbers are friendly (x² − 5x + 6 = (x−2)(x−3)), but the formula still works when they are not, including when Δ is negative.

This page also reports the vertex (−b/2a, f(−b/2a)), the axis of symmetry x = −b/2a, and Vieta’s formulas: the sum of roots is −b/a and the product is c/a, which remain true for complex roots. Calculation stays in your browser. Related tools: the slope calculator for a straight line instead of a parabola, the exponent calculator when you need a power rather than a root, and the scientific calculator for a one-off square root.

Using this ax² + bx + c solver:

1. Enter a, b, and c. a is the x² coefficient and cannot be 0. b and c may be zero or negative. The default is 1, −5, 6.
2. Read Δ. Positive: two distinct real roots. Zero: one repeated real root (the vertex is on the x-axis). Negative: two complex roots written as p ± qi.
3. Check the plugged-in formula, the vertex, the axis, and the Vieta sum and product against a homework key.
4. Copy the result block, or clear a, b, and c. Coefficients are saved locally for up to 30 days.

Note: this is IEEE-754 floating point, not a computer-algebra exact field. A discriminant that should be 0 but lands at 1e-18 will be treated as negative or positive according to its bit pattern. Very large |a|, |b|, |c| can overflow.

Derive it by dividing through by a (legal because a ≠ 0), moving c/a, and completing the square: x² + (b/a)x = −c/a, add (b/2a)² to both sides, then take square roots. The ± is not optional decoration: a genuine quadratic has two roots in ℂ, counted with multiplicity. When Δ = 0 those two roots are equal.

For a = 1, b = −5, c = 6: Δ = 25 − 24 = 1, √Δ = 1, so x = (5 ± 1)/2, which is 3 and 2. The result panel writes the substitution with your current a, b, c so a grader can see the arithmetic, not only the final pair.

The discriminant does not tell you whether the parabola opens up or down - that is the sign of a. It does not tell you the vertex height except in the Δ = 0 case, when the vertex y is 0. Graphing y = ax²+bx+c and solving ax²+bx+c = 0 are the same curve; the formula only finds the zeros.

If Δ < 0 then √Δ = i √(−Δ). The roots are

For real a, b, c they are conjugates. Example: x² + 1 = 0 has a = 1, b = 0, c = 1, Δ = −4, roots ± i. Example: x² + 2x + 5 = 0 has Δ = 4 − 20 = −16, real part −1, imaginary part 2, roots −1 ± 2i. This calculator writes that rectangular form. It does not switch to polar r∠θ unless you convert with a different tool.

Complex zeros still satisfy Vieta: sum −b/a and product c/a. They do not appear as x-intercepts on a real graph. If a course forbids complex numbers, a negative discriminant means “no real solution,” which is the same computation with the last step omitted.

Completing the square, or differentiating f(x) = ax²+bx+c and setting f′ = 2ax+b = 0, both give the axis x = −b/(2a). Plug that x back into f to get the vertex y-coordinate. For x² − 5x + 6 the axis is x = 2.5 and f(2.5) = 6.25 − 12.5 + 6 = −0.25, so the vertex is (2.5, −0.25). If a > 0 this is a minimum; if a < 0 a maximum.

The vertex is not a root unless Δ = 0. The default parabola dips below the x-axis between 2 and 3; the roots are the crossings, the vertex is the bottom of the valley.

If a = 0 the x² term is gone. You are left with bx + c = 0. When b ≠ 0 that is a line with one root x = −c/b. When b = 0 as well, either every x works (c = 0) or none does (c ≠ 0). The quadratic formula’s denominator 2a would be zero. This calculator therefore rejects a = 0 with an explicit message instead of returning Infinity or NaN. A linear solver is a different page.

x² − 5x + 6 = 0. Δ = 1, x = 3 and x = 2, vertex (2.5, −0.25), sum 5, product 6.

x² − 2x + 1 = 0 (that is (x−1)²). Δ = 0, x = 1 repeated, vertex (1, 0), sum 2, product 1.

x² + 1 = 0. Δ = −4, x = i and x = −i, vertex (0, 1), sum 0, product 1. 2x² + 3x − 2 = 0 has Δ = 9 + 16 = 25, x = (−3 ± 5)/4, so x = 1/2 and x = −2.

Completing the square is the derivation, not a rival answer. Divide by a, isolate the x terms, add (b/2a)², take square roots. The formula is that sequence compressed into one line. For x² − 5x + 6 you can also factor (x−2)(x−3) because 2 and 3 add to 5 and multiply to 6. Factoring fails on x² + 1 without complex numbers; the formula does not fail - it reports ±i.

Graphing calculators that “intersect y = 0” only see real roots. If Δ < 0 they show no intersection; this page still lists the complex pair. That is a feature for algebra II courses that include ℂ, and a reminder for courses that do not: “no real solution” is Δ < 0, not a crashed formula.

Plot the vertex first: x = −b/(2a), y = f of that x. Draw the axis as a dashed vertical line through the vertex. If Δ ≥ 0, mark the real root or roots on the x-axis (the default marks 2 and 3). Sketch the U through those three points if a > 0, or the n-shape if a < 0. The y-intercept is always (0, c) because f(0) = c. For x² − 5x + 6 that intercept is (0, 6), well above the vertex at (2.5, −0.25).

This calculator does not draw the sketch. It gives the numbers a graphing assignment needs. A slope of the tangent at a point is the derivative 2ax+b, which is a different question from the slope of a chord; use the slope calculator only for two sample points on the parabola if a teacher asked for a secant.

If the roots are r and s, then (x−r)(x−s) = x² − (r+s)x + rs. Matching ax²+bx+c after dividing by a gives r+s = −b/a and rs = c/a. For 2 and 3, sum 5 = −(−5)/1 and product 6 = 6/1. For ±i, sum 0 and product (−i)(i) wait: i · (−i) = −i² = 1, which equals c/a for x² + 1. The identities do not require the roots to be real. This page prints both as a check: if you copied x₁ and x₂ by hand, add and multiply them and compare.

Vieta is also how you build a quadratic from roots without the formula: x² − (sum)x + (product) = 0 when a = 1. Scaling by a puts the leading coefficient back.

Integer factoring hunts a pair that multiplies to a·c and adds to b. It is fast when it works and silent when it does not. The quadratic formula always returns two roots in ℂ (counted with multiplicity). Use factoring on quizzes that forbid calculators and have tiny integers. Use the formula when coefficients are ugly, when Δ is not a perfect square, or when you need the vertex in the same breath.

Completing the square is better when the task is to write vertex form y = a(x−h)² + k. The formula hides h = −b/(2a) and k = f(h) inside the algebra; this page surfaces them explicitly so you can write vertex form without re-deriving.

Subtracting two close numbers in −b ± √Δ can cancel digits when Δ ≈ b² (one root near 0, one near −b/a). Professional libraries use a numerically stable rearrangement. This homework calculator uses the textbook ± and IEEE doubles. For classroom a, b, c that is enough. For a = 1, b = 1e8, c = 1 you may see the small root look noisier than a CAS fraction.

Powers such as (2x)² belong on an exponent calculator or the scientific calculator before you collect a, b, c. Expanding (x−2)(x−3) by hand is still the right way to check x² − 5x + 6; the formula should then return 2 and 3, which it does on the default load.

Another classroom check: 2x² + 7x − 4 = 0 has Δ = 49 + 32 = 81, √Δ = 9, so x = (−7 ± 9)/4. That is x = 1/2 and x = −4. Sum −b/a = −7/2 = −3.5, and 0.5 + (−4) = −3.5. Product c/a = −2, and (1/2)(−4) = −2. If those Vieta checks fail, you copied a root with the wrong sign, not a new kind of quadratic.

Equations that are not already equal to zero must be moved first. x² = 5x − 6 is the default after subtracting 5x and adding 6. (x+1)² = 2 expands to x² + 2x + 1 − 2 = 0, so a = 1, b = 2, c = −1, Δ = 8, roots −1 ± √2. Typing the original squared binomial into a single box is not supported; this tool only takes the three coefficients of the standard form. A cubic such as x³ − 1 = 0 is also out of scope even though x = 1 is an obvious real root: the formula here is strictly degree two. Equations like 1/x + x = 3 are not quadratic until you multiply through by x (getting x² − 3x + 1 = 0, provided x ≠ 0). Do that algebra off-page, then type the resulting a, b, and c. Word problems that hide the quadratic (“a rectangle is 3 m longer than it is wide and the area is 40 m²”) still reduce to ax²+bx+c=0 after you name the unknown; this solver does not parse the English.

• Degree 2 only. No cubics, no quartics, no systems.
• Floating point, not fractions. 1/3 as a coefficient must be typed as a decimal or 0.333…. Roots that are exactly 2 and 3 will display as 2 and 3; messy irrationals are rounded to 12 significant figures.
• No step-by-step factoring. It does not search for integer factor pairs. It always uses the formula.
• Complex output is a+bi, not a polar pair, and not a plot.