Zeros of a Function Calculator
Enter the coefficients a, b and c to find the zeros of a linear or quadratic function.
The zeros of a function are the input values that make it equal zero — where its graph crosses the x-axis. For a quadratic ax² + bx + c, the zeros come from the quadratic formula, x = (−b ± √(b² − 4ac)) ÷ 2a. If the discriminant is negative, the zeros are a complex conjugate pair.
How to Use This Calculator
Write your function in the form ax² + bx + c and enter the three coefficients. For a linear function, set a to 0 and the calculator solves bx + c = 0 instead. The result is the zero or zeros — the x-values where the function equals zero.
The zeros are also called the roots or the x-intercepts, because they are where the graph crosses the x-axis. A quadratic can have two, one, or no real zeros.
How the Calculation Works
For a quadratic, the calculator uses the quadratic formula. The part under the square root, b² − 4ac, is the discriminant, and its sign decides the outcome: positive gives two distinct real zeros, zero gives one repeated real zero, and negative gives two complex conjugate zeros.
When a is zero the equation is linear, so the single zero is simply −c ÷ b. When the discriminant is negative, the calculator reports the complex roots in a ± bi form.
x = (−b ± √(b² − 4ac)) ÷ (2a)
Worked Example
For x² − 5x + 6, the coefficients are a = 1, b = −5, c = 6. The discriminant is (−5)² − 4(1)(6) = 25 − 24 = 1. Its square root is 1.
So x = (5 ± 1) ÷ 2, giving x = 3 and x = 2. The function has zeros at x = 2 and x = 3, which matches its factored form (x − 2)(x − 3).
Assumptions and Limitations
- Handles linear (a = 0) and quadratic functions
- Reports real zeros, a repeated zero, or a complex conjugate pair
- For higher-degree polynomials, a different method is needed
- Results are rounded for display
Who Uses a Zeros of a Function Calculator
Algebra and precalculus students use this to find the zeros, or roots, of a linear or quadratic function — the x-values where it equals zero. It supports graphing, factoring practice, and homework checking. Enter the function to get its roots, which are also where the graph crosses the x-axis.
Related Calculators
Discriminant Calculator
Find b² − 4ac and what it tells you about a quadratic's roots.
Open calculator →Simplify Radical Calculator
Reduce a square root to its simplest radical form, step by step.
Open calculator →Complex Number Calculator
Add, subtract, multiply or divide complex numbers with steps.
Open calculator →Elimination Calculator (2×2 System)
Solve a system of two linear equations by elimination.
Open calculator →Frequently Asked Questions
What are the zeros of a function?
They are the input values that make the function equal zero — the points where its graph crosses the x-axis. They are also called roots or x-intercepts.
How do you find the zeros of a quadratic?
Use the quadratic formula, x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant b² − 4ac tells you whether there are two real zeros, one repeated zero, or two complex zeros.
Can a function have no real zeros?
Yes. If the discriminant is negative, the quadratic never crosses the x-axis and its zeros are a complex conjugate pair rather than real numbers.
What is the difference between zeros and the discriminant?
The discriminant, b² − 4ac, tells you the nature and number of zeros. The zeros are the actual x-values. This tool computes both — the discriminant along the way and the zeros as the result.
How do I find the zero of a linear function?
Set a to 0 so the function is bx + c. The single zero is −c ÷ b, the x-value where the straight line crosses the x-axis.
Looking for a different calculator?
CalculatorPlus has free tools across finance, construction, math, health and more — each one showing the formula and a worked example.