Zeros of Quadratic Polynomial

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 in 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. 
 

Polynomials of degree 2 are known as quadratic polynomials. The general form of a quadratic polynomial is ax2 + bx + c, where a ≠ 0, a and b are the coefficients, x is the variable, and c is the constant. 

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: 

• Factorization Method
• 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. For example, find the zeros of a quadratic polynomial x2 - 7x + 12 = 0
Step 1: Arrange the polynomial in standard form (ax2 + bx + c = 0)
Here, the given polynomial is in standard form: x2 - 7x + 12 = 0
Here, a = 1, b = -7, and c = 12

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

Step 3: Splitting the middle term 
Here, we split 7x as 3x and 4x, then the quadratic polynomial becomes:
x2 - 3x - 4x + 12 = 0

Step 4: Grouping and factoring 
Now we factor, x2 - 3x - 4x + 12 = 0 as:
(x2 - 3x) - (4x - 12) = 0
x(x - 3) - 4(x - 3) = 0
(x - 3)(x - 4) = 0

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: ax2 + bx + c = 0 is: x = -b ± b2 - 4ac2a.
For example, find the zeros of a quadratic polynomial 3x2 - 5x + 2 = 0
Here, a = 3
b = -5
c = 2

Substituting the values in the formula: x = -b ± b2 - 4ac2a
x = - -5 ± (-5)2 - 4 × 3 × 22 × 3
x = 5 ± 25 - 246
x = 5 ± 16
x = 5 ± 16
x = 5 + 16 and x = 5 - 16
x = 6/6 = 1
x = 4/6 = ⅔

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

The nature of zeros of a quadratic polynomial refers to the nature of zeros, that is, whether the zeros are real, distinct, or complex numbers. The nature of zeros is determined using the discriminant(D) of the quadratic polynomial: D = b2 - 4ac. 
 

The sum and product of the zeros of a quadratic polynomial show the relationship between its coefficients and roots of a quadratic equation. Let α and β be the zeros of the quadratic polynomial ax2 + bx + c = 0. 
The sum of zeros: α + β = -b/a 
The product of zeros: αβ = c/a
 

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, and they represent the real 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.

• 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. 
• 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. 
• 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.
• 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.