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Last updated on September 22, 2025
In an equation, values that make the function equal to 0 are called the zeros of the function. In this article, we will learn what zeros of a function are, methods to find them, how they appear on a graph, and the formulas involved
The zeros of a function f(x) are the values of x that make f(x) = 0. The values of x are known as the roots of a function. Graphically, the zeros are the points where the function crosses or touches the x-axis, also called x-intercepts.
There are numerous methods to find the zeros of a function. These different methods include:
For example: f(x) = x2 + 7x + 10
It can be factorized into: f(x) = (x + 2)(x + 5)
Setting each factor equal to zero: x + 2 = 0 and x + 5 = 0
Finding the value of x:
x + 2 = 0
x = -2
x + 5 = 0
x = -5
Therefore, the zeros of the function f(x) = x2 + 7x + 10 are:
x = -2 and x = -5.
x = [-b ± √(b² - 4ac)] / 2a
Assume x1 = x0 + h, where x0 is the approximate root of the equation
f(x1) = 0, so f(x0 + h) = 0
Using Taylor’s theorem, expanding f(x0 + h):
f(x0) + hf′(x0) + …. = 0
Then h = -fx0f'x0
So, x₁ = x₀ - f(x₀)/f'(x₀)
xn + 1 = xn - fxnf'xn
Assume the points a and b such that a < b and f(a)∙f(b) < 0.
Next, calculate the midpoint of a and b, so m = (a + b)/2
The next interval is selected based on the sign of f(m), that is:
If f(m) = 0, then m is the root
If f(m) < 0, choose the interval from m to b
If f(m) > 0, choose the interval from a to m.
The zeros of a function f(x) represent the solutions to the equation: f(x) = 0. In other words, we are finding the value of x, which makes f(x) = 0. For the value of x, different methods like grouping, algebraic identities, splitting the middle term, etc., are used.
How to Find Real Zeros of a Function?
The real zeros of a function f(x) are real numbers r that make f(x) = 0. In other words, it is a value of x for which the function equals zero.
For example, f(x) = 3x3 - 6x2 - 9x
= 3x(x2 - 2x - 3)
Factoring x2 - 2x - 3 as (x - 3)(x + 1)
So, f(x) = 3x(x - 3)(x + 1)
3x(x - 3)(x + 1) = 0
x - 3 = 0 → x = 3
x + 1 = 0 → x = -1
x = 0
So, the real zeros of the function f(x) = 3x3 - 6x2 - 9x are x = 0, x = 3, and x = -1.
How to Represent Zeros of a Function on a Graph
We find the zeros of a function using a graph; that is, the point where the graph intersects the x-axis is the root of the function. Here, we will learn how to find the value of the root of f(x2 - 4) using the graph.
The graph is of the function: f(x) = x2 - 4. Here, the graph intersects the x-axis at two points (2 and -2), so x = 2 and x = -2. Therefore, 2 and -2 are the roots (zeros) of the function that make f(x) = 0.
The zeros of a function are used in real life in the fields of physics, engineering, computer security, mathematics, etc. In this section, we will learn the applications of the zeros of a function.
When finding the zeros of a function f(x) = 0, students often make mistakes. In this section, we will learn some common mistakes and ways to avoid them in the zeros of a function.
Find the zero of a function: 2x - 6
x = 3
Finding the value of x in 2x - 6
2x - 6 = 0
2x = 6
x = 3
Find the root of x2 - 5x + 6
x = 2 or x = 3
Factoring the quadratic equation:
x2 - 5x + 6 = (x - 2)(x - 3)
x - 2 = 0 and x - 3 = 0
Solving the equations:
x - 2 = 0
x = 2
x - 3 = 0
x = 3
Here, the value of x is 2 and 3.
Find the root of x2 + 4x + 4
Here, x = -2
Using the quadratic formula to find the zeros of the function:
x = -b ± b2 - 4ac2a
Here, a = 1, b = 4, and c = 4
x = -4 ± 42 - 4 × 1 × 42 × 1
x = [-4 ± √(16 - 16)] / 2 = -2
Find the root of -4x + 8
x = 2
Setting the equation equal to zero to find the value of x:
-4x + 8 = 0
-4x = -8
x = -8/-4
x = 2
Find the root of 3x3 - 6x2 + 9x
x = (2 ± √2i)
To find the root of 3x3 - 6x2 + 9x, we first factor out the equation
f(x) = 3x3 - 6x2 + 9x
f(x) = 3x(x2 - 2x + 3)
That is 3x = 0 and x2 - 2x + 3 = 0
Solving the equation:
3x = 0 ⇒ x = 0
x2 - 2x + 3 = 0
Finding the value of x using the quadratic formula:
x = [-b ± √(b² - 4ac)] / 2a
Here, a = 1, b = -2, c = 3
x = [2 ± √(4 - 12)]/2 = [2 ± √(-8)]/2 = 1 ± √2i
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
: He loves to play the quiz with kids through algebra to make kids love it.