Quadratic Polynomial

A quadratic polynomial is a second-degree polynomial that has the highest exponent of 2. It is written in the form ax² + bx + c, where a ≠ 0. Here, x is the variable, a and b are coefficients, and c is the constant. 

The values of x that satisfy the equation are the roots of the quadratic polynomial.

The standard form of the quadratic polynomial is ax2 + bx + c = 0. There are different methods to find the solution to a quadratic equation:

• Factorizing 
   
• Completing the square
   
• Quadratic formula
   
• Graph 

The quadratic formula is the simplest way to find the roots of a quadratic polynomial equation. To find the solution of a quadratic equation, we use the quadratic formula:

x = -b  ± √b² - 4ac/2a

The expression under the square root b²-4ac is the discriminant of the root. The discriminant D is used to determine the nature of the root.

The value of x in a quadratic equation is the root of a quadratic polynomial; it is also known as a solution or zero. 

To analyze the nature of the root, we use the discriminant, D = b² - 4ac, the root can be both real and imaginary. Based on the discriminant, the roots can be:

• If D > 0, then the roots are real 
   
• If D < 0, then the roots are imaginary
   
• If D = 0, then the root has one real and repeated root. 

To understand the relationship between the roots and the coefficients, we use the roots of a quadratic equation. The sum and product of the roots of a quadratic equation in standard form are calculated using the coefficients. Let the roots be α and β, for a quadratic equation: ax2 + bx + c = 0, then:

• Sum of the roots: α + β = -coefficient of x / coefficient of x² = -b/a

• Product of the roots: α∙β = constant / coefficient of x² = c/a

If the sum and product of the roots are given, the quadratic equation can be obtained by:

x² - (α + β)x + α∙β = 0 

For example, find the sum and product of the roots of x² - 8x + 15 = 0

Here, a = 1, b = -8, and c = 15

Sum of the roots = -b/a 

= -(-8)/1 = 8

Product of the roots: c/a 

= 15/1 

= 15

The graph of the quadratic polynomial forms a parabola, which may open upwards or downwards depending on the value of a. The quadratic equation is in the form of ax² + bx + c. The equation of the parabola is y = ax² + bx + c. To plot the graph, the value of x is substituted into the equation to find the value of y, resulting in the points (x, y) on the curve. The points where the parabola intersects the x-axis represent the real roots of the equation, if they exist. If the value of the discriminant is negative, then the graph does not cross the x-axis, showing that the equation has no real roots.

To form the quadratic polynomial, we use the values of the roots of the equation. The polynomial can be formed using the formula: x2 - (α + β)x + α.β = 0, where α and β are the roots of the quadratic equation.  

For example, if the roots of a quadratic polynomial are 5 and 3, find the quadratic polynomial 
The general form of a quadratic polynomial with roots is x2 - (α + β)x + α  β = 0 
Here, α = 5
β = 3 
That is x2 - (5 + 3)x + 5 × 3 = 0 
x2 - 8x + 15 = 0

Factorizing the quadratic polynomial is a method used to find the roots of the quadratic equation. In this process, the polynomial is broken down into a product of factors, which is reverse multiplication. A few factorization methods are:

• Common Factor Method
   
• Sum of Difference Method
   
• Factor by Grouping Method
   
• Perfect Square Trinomials Method

If all the terms in a polynomial share a common factor, we can factor it out. By using the distributive law in reverse, that is 
x(a + b) = xa + xb. 

Factoring out x, as it is the common factor: 
xa + xb = x(a + b). 

For example, factoring 4x2 + 2x = 0

Here, the common factor is 2x, factoring it out

4x2 + 2x = 2x(2x + 1)

So, it can be factorized as 2x(2x + 1)

When the two factors of a polynomial are the same, but one is addition and the other is subtraction, we use the sum of difference method to factorize the polynomial. This can be represented as: (x - a)(x + a)
It can be expanded as: 
(x - a)(x + a) = x2 - ax + xb - a2
= x2 - a2 

Find the solution of (x + 8)(x - 8) using the sum of the difference method

Using the identity: (a + b)(a - b) = a2 - b2
Here, a = x and b = 8
So, (x + 8)(x - 8) = x2 - 82
= x2 - 64

In the grouping method, the terms of a polynomial are arranged into pairs or groups such that each group has a common factor. To factor the polynomial by the grouping method, follow these steps:
Factor out the common factor from the quadratic polynomial
Each group of the expression is factorized
Now factoring the common binomial

For example, factoring x2 + 5x + 2x + 10 by grouping method
Grouping the terms: 
x2 + 5x + 2x + 10 = (x2 + 5x) + (2x + 10)
In x2 + 5x, the common factor is x
So, x(x + 5)
The common factor in (2x + 10) is 2
So, 2(x + 5)

x(x + 5) + 2(x + 5) = (x + 5)(x + 2)

Factoring out the common binomial, (x + 5)(x + 2)

The equations with the pattern of a perfect square are solved using the perfect square trinomial method 
The perfect square trinomial formulas are:
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2

For example, factor the quadratic equation x2 + 16x +64
Here, a2 = x2, a = x
2ab = 16x, ab = 8x
b2 = 64, so b = 8

x2 + 16x + 64 = x2 + (2 × x × 8) + 82
= (x + 8)2

In real life, we use the quadratic polynomial in different fields like sports, physics, finance, engineering, etc. Here are some of the applications of quadratic polynomials.

• In sports like basketball, soccer, disc throw, javelin throw, and shot put, the motion of the ball or the object will be in a parabolic path. We use the quadratic equation to calculate the maximum height, modeling projectile motion, and to determine the distance traveled. 

• For the natural movements of characters and objects in video games and animation, the quadratic polynomials are used.

• In business, to model and solve problems related to profit maximization, optimal production level, and cost minimization.