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 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. 
 

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: 

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 
 

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.

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
   

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. 
   

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.
   

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.