Zero Calculator
This Zero Calculator finds the zeros of a polynomial: the values of x that make the equation equal to zero, also called roots or x-intercepts. Pick a degree, linear (ax + b = 0), quadratic (ax² + bx + c = 0) or cubic (ax³ + bx² + cx + d = 0), enter the coefficients, and it solves for x, including the complex pair when a quadratic has no real zeros. It works from coefficients rather than a typed-out equation, so there is no expression syntax to get wrong.
Cubic mode reports real zeros only, found exactly with Cardano’s method rather than by sampling the curve. Quadratic mode also reports the complex pair when the discriminant is negative. Results are shown to four decimal places.
Quick Answer
A zero of a function is an x value that makes the function equal to 0. For the pre-filled linear example, 2x – 6 = 0, the zero is x = 3.0000. Switch to quadratic and the pre-filled x² – 5x + 6 = 0 gives two zeros, x = 3.0000 and x = 2.0000; the pre-filled cubic x³ – 6x² + 11x – 6 = 0 gives three, x = 1.0000, 2.0000 and 3.0000.
- Quick Answer
- How to use the zero calculator
- The method behind each degree
- Worked examples in all three modes
- Zeros, roots and x-intercepts: three names, one idea, with a graph
- What competing zero calculators miss
- The limits of this tool, stated plainly
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the zero calculator
Choose the degree that matches your equation, type the coefficients, and click Calculate. The zeros appear under the equation the tool echoes back to you.
- Pick the function type. Linear for degree 1, quadratic for degree 2, cubic for degree 3. The highest power of x in your equation is the degree.
- Enter every coefficient, including zeros. If a term is missing, its coefficient is 0: x² – 25 has b = 0, and x³ – 8 has b = 0 and c = 0. Leaving a field blank is treated as missing input, not as zero, and the calculator asks for valid coefficients instead.
- Mind the sign. Coefficients carry their signs with them: in 2x – 6, b is -6, not 6. Most wrong answers from this kind of tool are sign slips, not method errors.
- Read the zeros list. Each line is one zero, to four decimal places. A quadratic with a negative discriminant lists its two complex zeros instead.
If your equation is written with the terms in a different order, or with everything on the right-hand side, rearrange it into the standard form first: all terms on the left, set equal to zero, highest power first. The coefficients are then simply the numbers in front of each power, as covered in any polynomial calculator that works with polynomial arithmetic.
The method behind each degree
Each degree is solved with its own exact method, not one generic routine. Linear and quadratic modes use the closed formulas; cubic mode uses Cardano’s method on the depressed cubic, the standard exact solution catalogued in NIST’s Digital Library of Mathematical Functions, section 4.43.
| Degree | Form | Method used here | Possible zeros |
|---|---|---|---|
| Linear | ax + b = 0 | x = -b / a | Exactly 1 real zero |
| Quadratic | ax² + bx + c = 0 | Quadratic formula, with the discriminant b² – 4ac deciding the case | 2 real, 1 repeated, or 2 complex |
| Cubic | ax³ + bx² + cx + d = 0 | Cardano’s method: depress the cubic (x = t – b/3a), then the trigonometric or cube-root case | 3 real (some may repeat) or 1 real; this tool reports the real zeros |
Linear: one step
The zero of ax + b is x = -b / a, a single rearrangement. For 2x – 6, x = 6 / 2 = 3. If you want the working for linear equations in general, a three-step linear equations calculator shows the isolate-the-variable routine this formula compresses.
Quadratic: the discriminant decides
The quadratic formula, x = (-b ± √(b² – 4ac)) / 2a, covers every case, and the discriminant b² – 4ac tells you which case you are in before you finish: positive means two distinct real zeros, zero means one repeated zero, negative means two complex zeros, which this calculator reports as a conjugate pair such as 0.0000 + 1.0000i and 0.0000 – 1.0000i for x² + 1. For a deeper treatment of the formula itself, the quadratic equation calculator is dedicated to this one degree.
Cubic: Cardano, not curve sampling
Cubic mode divides through by a, substitutes x = t – b/(3a) to remove the square term, and solves the depressed cubic t³ + pt + q exactly: the trigonometric solution when three real zeros exist, cube roots (the same cube roots a cube root calculator computes) when only one does. That matters because the obvious shortcut, testing x values on a grid and reporting the near misses, silently fails on ordinary cubics: it finds no zero at all for x³ – 2 (true zero 1.2599, which falls between grid points) and none for 100x³ – 103.0301 (true zero 1.0100, where the curve is too steep for a sampling tolerance). An exact method has no grid to fall through.
Worked examples in all three modes
The three pre-filled modes are the worked examples, and each reproduces exactly. Linear: 2x – 6 = 0 gives x = -(-6)/2 = 3.0000. Quadratic: x² – 5x + 6 = 0 has discriminant 25 – 24 = 1, so x = (5 + 1)/2 = 3.0000 and x = (5 – 1)/2 = 2.0000. Cubic: x³ – 6x² + 11x – 6 = 0 factors as (x – 1)(x – 2)(x – 3), and Cardano’s method returns the same three zeros, 1.0000, 2.0000, 3.0000.
One display quirk to know before you check these by hand: the Function line echoes the coefficients literally, so the linear example prints as “2x + -6 = 0” rather than “2x – 6 = 0”. The arithmetic uses the signed values correctly either way; the echo is a formatting habit of the tool, kept so the numbers on screen are exactly the numbers you typed.
More runs of the same tool, including the edge cases:
| Equation entered | Mode | Zeros returned |
|---|---|---|
| 3x + 1 = 0 | Linear | x1 = -0.3333 |
| x² – 25 = 0 | Quadratic | x1 = 5.0000, x2 = -5.0000 |
| x² – 4x + 4 = 0 | Quadratic | x1 = 2.0000 (repeated zero, listed once) |
| x² + 1 = 0 | Quadratic | x1 = 0.0000 + 1.0000i, x2 = 0.0000 – 1.0000i |
| x³ – 8 = 0 | Cubic | x1 = 2.0000 |
| x³ – 2 = 0 | Cubic | x1 = 1.2599 |
| 100x³ – 103.0301 = 0 | Cubic | x1 = 1.0100 |
Zeros, roots and x-intercepts: three names, one idea, with a graph
A zero, a root and an x-intercept describe the same points from three directions: a zero of the function, a root of the equation f(x) = 0, and the place the graph crosses or touches the x-axis. OpenStax’s Algebra and Trigonometry (§5.5, Zeros of Polynomial Functions) treats them together for this reason, and adds the machinery used to hunt them by hand: the Factor Theorem and the Rational Zero Theorem.
The graph below is the pre-filled quadratic, x² – 5x + 6. It crosses the axis at exactly its two zeros, x = 2 and x = 3:
Multiplicity is the one refinement worth carrying away. A zero that appears twice, like x = 2 in x² – 4x + 4 = (x – 2)², does not cross the axis; the parabola touches it and turns around. This calculator lists a repeated zero once, so a single line in the results for a quadratic means a repeated zero, not a missing one. NIST’s DLMF §3.8 (Nonlinear Equations) defines multiplicity in exactly these terms: a zero of multiplicity m is where the function and its first m – 1 derivatives all vanish.
What competing zero calculators miss
The big general solvers are powerful, and for a quick coefficient job like this one their power is mostly friction:
- Free-form equation input invites silent misreads. Symbolab and WolframAlpha accept typed or pasted expressions, which is flexible until an implicit multiplication or a misread exponent changes the equation. Their step-by-step working is also partly paywalled (Symbolab’s full steps sit behind its subscription). Coefficient fields cannot misread you: the numbers you typed are the numbers echoed back.
- Graph-first tools approximate. Graphing calculators such as Desmos show intercepts visually, and grid-sampling root finders report near misses. Both are estimation habits; on x³ – 2 a sampling grid finds nothing, where the exact cubic solution is 1.2599. This page solves each degree with its closed method instead.
- Degree specialists stop at their degree. The strong polynomial-roots tools (Firgelli’s covers degrees 1 to 4 with Cardano and Ferrari) are engineering pages with long formula dumps around a small form, and single-degree pages handle only their own case. Here the three school and undergraduate degrees sit in one form with one results format, and the honest boundary is stated plainly: degree 4 and above are not covered.
- Complex zeros are often dropped. Several zeros calculators report only real roots and call x² + 1 unsolvable. Quadratic mode here returns the conjugate pair, because “no real zeros” and “no zeros” are different statements.
The limits of this tool, stated plainly
Four boundaries, so nothing on this page oversells the form above:
- Degrees 1 to 3 only. Quartics and higher polynomials are not solved here. A general polynomial calculator handles polynomial arithmetic, but root-finding for degree 4 and above belongs to a dedicated solver.
- Cubic mode reports real zeros only. When a cubic has one real zero, the other two are a complex pair, and this tool does not display them. Quadratic mode does display its complex pair.
- Coefficients, not expressions. There is no typed-equation parser, no fractions button and no variable other than x. That is the trade for never misreading an expression.
- Four decimal places. Zeros are rounded for display: -0.3333 for 3x + 1 is a rounding of -1/3, not the exact value. For homework that asks for exact form, state the fraction or surd alongside the decimal.
Frequently asked questions
A zero is an input value x that makes the function output 0. For f(x) = 2x – 6, the zero is x = 3, because f(3) = 0. On a graph, the real zeros are the x-coordinates where the curve crosses or touches the x-axis.
They name the same values from different angles: a zero of the function, a root of the equation f(x) = 0, and an x-intercept of the graph. Complex zeros are the exception: they are genuine roots, but they never appear as intercepts on a real graph.
Because a multiplies the highest power of x, and if a = 0 that term disappears and the equation drops a degree: a “quadratic” with a = 0 is really a linear equation. The calculator stops and tells you, rather than silently solving a different equation.
The quadratic has no real zeros, and its graph never touches the x-axis. It still has two complex zeros, a conjugate pair, and this calculator reports them, for example x² + 1 gives 0.0000 + 1.0000i and 0.0000 – 1.0000i.
Yes. A cubic’s ends run to opposite infinities, so its graph must cross the x-axis at least once. A cubic has either three real zeros (counting repeats) or one real zero and a complex pair; this tool reports the real ones.
With Cardano’s method, the exact algebraic solution: depress the cubic, then apply the trigonometric form when there are three real zeros or cube roots when there is one. It does not sample the curve on a grid, so it cannot step over a zero the way sampling can.
The Function line echoes the coefficients exactly as entered, signs included, so 2x – 6 prints as “2x + -6 = 0”. It is a display habit, not a different equation: the calculation uses the signed coefficients, and the zero is still x = 3.
No. It covers degrees 1, 2 and 3 only: linear, quadratic and cubic. For degree 4 and above, or for zeros of non-polynomial functions such as trigonometric or exponential equations, you need a general equation solver or a graphing tool.
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References
- OpenStax, Rice University. Algebra and Trigonometry 2e, §5.5: Zeros of Polynomial Functions. Accessed October 24, 2026.
- OpenStax, Rice University. Precalculus 2e, §3.6: Zeros of Polynomial Functions. Accessed October 24, 2026.
- National Institute of Standards and Technology. Digital Library of Mathematical Functions, §4.43: Cubic Equations. Accessed October 24, 2026.
- National Institute of Standards and Technology. Digital Library of Mathematical Functions, §3.8: Nonlinear Equations (zeros and multiplicity). Accessed October 24, 2026.
- National Institute of Standards and Technology, Information Technology Laboratory. Dataplot tutorial: finding roots of an equation over an interval. Accessed October 24, 2026.