Zeros of Quadratic Polynomial
2026-02-28 08:43 Diff
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Last updated on October 30, 2025

Zeros of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, they are the solutions to the equation f(x) = 0. In this article, we will learn about the zeros of a quadratic polynomial and the methods to find them, such as factorization and the quadratic formula.

What are Zeros of Quadratic Polynomials?

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The value of x that makes a quadratic polynomial equal to zero is known as the zero of a quadratic polynomial.

For a quadratic polynomial of the form f(x) = ax2 + bx + c, where a ≠ 0, the zeros of the quadratic polynomial are the values x for which f(x) = 0.

The zeros of a quadratic polynomial are also known as roots or solutions of the polynomial. 
 

What is a Quadratic Polynomial?

How to find Zeros of Quadratic Polynomials?

A quadratic polynomial can have a maximum of two zeros because its highest degree is 2. Now, let’s learn how to find the zeros of a quadratic polynomial.

The common methods used to find the zeros of a quadratic polynomial are: 

  1. Factorization Method
  2. Quadratic Formula
  • Factorization Method
    In the factorization method, to find the zeros of a quadratic polynomial, we set the equation as the product of two linear polynomials by factoring them.

Let's practice. 

Find the zeros of a quadratic polynomial x2 - 7x + 12 = 0

  1. Step 1: Arrange the polynomial in standard form (ax2 + bx + c = 0)
    Standard form: x2 - 7x + 12 = 0
    Here, a = 1, b = -7, and c = 12
     
  2. Step 2: Finding two numbers
    Now find two numbers whose product is ac and sum is b
    Here, the numbers are -3 and -4, as -3 × -4 = 12 and -3 + -4 = -7
     
  3. Step 3: Splitting the middle term 
    Here, we split 7x as 3x and 4x, then the quadratic polynomial becomes:
    x​​​​​​-2 - 3x - 4x + 12 = 0
     
  4. Step 4: Grouping and factoring 
    Now we factor, x​​​​​2 - 3x - 4x + 12 = 0 as:
    \((x^2 - 3x) - (4x - 12) = 0\\ x(x - 3) - 4(x - 3) = 0\\ (x - 3)(x - 4) = 0\)
     
  5. Step 5: Solve the equation 
    To find the value of x, we solve the equation:
    \(x - 3 = 0 → x = 3\\ x - 4 = 0 → x = 4\)
  • Quadratic Formula
    The quadratic formula to find the value of zeros of a quadratic polynomial in the form: ax​​​​​​2 + bx + c = 0 is:

Let's practice. 

Find the zeros of a quadratic polynomial 3x2 - 5x + 2 = 0
 

  1. Here, a = 3
    b = -5
    c = 2
     
  2. Substituting the values in the formula: \(x = \frac{-b \ ± \ \sqrt{b^2 - 4ac}}{2a}\)

    \(x = \frac{- (-5) \ ± \ \sqrt{(-5)^2 \ -\ 4 × 3 × 2}}{2 × 3}\\ x = \frac{5 \ ± \ \sqrt{25 - 24}}{6}\\ x = {5 ± 1\over 6}\\ x = {5+1\over 6} \ and \ x = {5-1\over 6}\\ x = {6\over 6} = 1\\ x = {4\over 6} = ⅔\)

The zeros of the quadratic polynomial are 1 and 2/3 
 

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Nature of Zeros of Quadratic Polynomial

The nature of zeros of a quadratic polynomial refers to the nature of zeros. It tells whether the zeros are real, distinct, or complex numbers.

The nature of zeros is determined using the discriminant(D) of the quadratic polynomial: D = b​​​2 - 4ac. 

The following table shows the nature of zeros according to the value of discriminant.

Value of discriminant 

Nature of zeros

If D < 0

The zeros are complex

If D > 0 

The zeros are real and distinct 

If D = 0

The zeros are real and equal 

Sum and product of Zeros of Quadratic Polynomial

The sum and product of the zeros of a quadratic polynomial show the relationship between its coefficients and roots of quadratic equation.

Let α and β be the zeros of the quadratic equation ax2 + bx + c = 0. 

  • The sum of zeros: α + β = -b/a 
  • The product of zeros: αβ = c/a
     

Graphical Representation of Zeros of Quadratic Polynomial

Let’s now learn how to find the zeros of a quadratic polynomial using a graph.

  • The graph of a quadratic polynomial forms a parabola.
     
  • The x-intercepts of this graph are the points where the graph intersects the x-axis.
     
  • The point of intersection represents the real zeros of a quadratic polynomial. 
     

Tips and Tricks to Master Zeros of Quadratic Polynomial

Zeroes of quadratic polynomials can be difficult for students of smaller grades. So, here are some essential tips and tricks to make it easy:

  1. To recognize if the given equation is a quadratic, check its degree.
     
  2. When factorizing, ensure that the signs in the equation do not change.
     
  3. If c = 0, then the zeroes of equation ax² + bx is x = 0 and \(x = { -b\over a}\)
     
  4. When finding factors seems difficult, directly use the formula
     
  5. Memorize the formula of discriminant, \(D = b^2 - 4ac\), to find the nature of roots or zeroes.

Parent Tips:

  • Help your child memorize all formulas.
  • You can check your child’s answer manually using the sum and product of the zeros’ formula. 
  • You can also use polynomial equation solver calculator to verify the zeros.
     

Real-World Applications of Zeros of a Quadratic Polynomial

The zeros of a quadratic polynomial are used in fields like physics, engineering, architecture, finance, etc. We will explore how the quadratic formula is applied in real-life applications.

  1. In physics, the zeros of a quadratic polynomial are used to understand the projectile motion. For example, to find the height of a ball if it is thrown upwards, we use the quadratic polynomial. 
     
  2. In structural design, a quadratic polynomial is used to model the parabolic arches or support structures. The zeros help us model a structure, like a beam or bridge, where there is no movement or where the stress changes. 
     
  3. In finance, the zeros of a quadratic polynomial are used to find the time of repayment of a loan or to calculate the number of payments needed.
     
  4. To describe the population growth or decline, we use the quadratic polynomial. The zeros of a quadratic polynomial help identify when the population might start or stop growing, or when it drops to a critical low point. 
     
  5. In video games, animations, VR simulations, and 3D modeling, designers and programmers use quadratic equations to model jumping, falling, throwing or projectile movements. Zeroes determine when an object hits the ground or returns to the original point.

Common Mistakes and How to Avoid Them in Zeros of Quadratic Polynomial

When finding the zeros of a quadratic polynomial, students make errors. Here are some common mistakes students make when finding the zeros of a quadratic polynomial, and the tips to avoid them. 
 

Problem 1

Find the zeros of a quadratic polynomial: x2 -5x + 6

Okay, lets begin

Here, x = 2 and x = 3
 

Explanation

  1. To find the zeros of the polynomial x2 - 5x + 6, we factor the polynomial
    x2 - 5x + 6 = (x - 2)(x - 3)
     
  2. The factors used are -2 and -3 because their product is 6 (which equals ac) and their sum is -5 (which equals b)
     
  3. Solving the equations: 
    x - 2 = 0 → x = 2
    x - 3 = 0 → x = 3 

Well explained 👍

Problem 2

Find the zeros of x2 - 4x + 4

Okay, lets begin

The polynomial has one real zero, x = 2. 
 

Explanation

  1. To find the root of the polynomial x2 - 4x + 4, we use the factorization method.  
    x2 - 4x + 4 = (x - 2)2
     
  2. So, x - 2 = 0
    x = 2

Well explained 👍

Problem 3

Find the sum and product of the quadratic polynomial: x2 + 2x + 5

Okay, lets begin

 The sum of the quadratic polynomial is -2, and the product of the quadratic polynomial is 5 
 

Explanation

  1. Here, a = 1, b = 2, and c = 5
     
  2. To find the sum of its zeros, we use the formula: -b/a
    -b/a = -2/1 = -2
     
  3. To find the product of the zeroes, we use the formula: c/a
    c/a = 5/1 = 5

Well explained 👍

Problem 4

Find the zeros of x2 + x - 6

Okay, lets begin

 x = 2 and x = -3 
 

Explanation

To find the zeros of a quadratic polynomial, we use the formula: \(x = \frac{-b \ ± \ \sqrt{b^2 - 4ac}}{2a}\)
Here, a = 1, b = 1 c = -6

\(x = \frac{-1 \ ± \ \sqrt {1^2 \ - \ 4(1)(-6)}}{2a}\)
\(x = {-1 \ ± \sqrt{1\ + 24} \over 2}\)
\(x = {-1 \ ± \ \sqrt{25} \over 2}\)

So, \(x = {-1 \ + \ 5 \over 2}\) and\( x = {-1 \ - \ 5 \over 2}\)

x = 4/2 = 2
x =  -6/2 = -3

Well explained 👍

Problem 5

Find the zeros of x2 + 6x + 9 and verify the relationship between zeros and coefficients of the polynomial.

Okay, lets begin

 x = -3 and x = -3. The relationship between zeros and coefficients holds. 
 

Explanation

To find the zeros of a quadratic polynomial, we use the formulas: \(x = \frac{-b \ ± \ \sqrt{b^2 - 4ac}}{2a}\)
Where, a = 1, b = 6, c = 9

 

\(x = \frac{-6 \ ± \ \sqrt{6^2 - 4(1)(9)}}{2(1)}\)
\(x = \frac{-6 \ ± \ \sqrt{36 - 36}}{2(1)}\)
\(x = \frac{-6 \ ± \ 0}{2(1)}\)
x = -6/2 = -3

So, the value of zeros are x = -3 and x = -3

  • The sum of zeros = -3 + -3 = -6
  • The product of zeros = -3 × -3 = 9

To verify the relationship between zeros and coefficients, for a quadratic polynomial ax2 + bx + c: 

  • Sum of zeros = -b/a = -6/1 = -6
  • Product of zeros = c/a = 9/1 = 9


So, the relationship between zeros and coefficients of the polynomial is verified. 
 

Well explained 👍

FAQs on Zeros of Quadratic Polynomial

1.What is the way my child can verify if the calculated value of zeroes are correct?

To check if the zeroes are correct, children can use the formula of sum and product of zeroes. 

  • Sum of zeroes = \(-\frac{b}{a} \)
  • Product of zeroes =  \(\frac{c}{a}\)

2.Why is it necessary for my child to learn about zeroes?

Because it will help your child to:

  • Connect algebra and geometry (zeroes correspond to where the parabola cuts the x-axis)
  • Solve real-life problems like finding time, distance, or profit when something “reaches zero”
  • Build a strong foundation for advanced-level algebra, calculus, and physics
     

3.How can parents help children visualize zeroes?

With the help of a graph, show how the parabola touches the x-axis, and the points are the zeroes.

You can also use examples like: “At what time will the height of a ball be zero again after being thrown?” — that’s finding a zero!
 

4.What common mistakes can my child make?

Here are a few mistakes

  1. Misinterpreting zeroes as coefficients or constants
  2. Incorrectly using formula
  3. Not considering the cases when some quadratics have no real zeroes
  4. Not verifying their answers.
     

5.How can I relate this to real life?

You can use examples such as:

  • When a ball thrown from a height hits the ground, and the height becomes zero.
  • When profit = 0, the company breaks even.
  • When the distance left to cover equals zero, the journey ends.
  • Each of these points denotes a zero, which is a point where something reaches nothing!