# Relation between Coefficients and Roots of Quadratic Equations.

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## Relation between Coefficients and Roots of Quadratic Equations.

Introduction:

In an algebraic term a numerical value or symbol which is multiplied by a variable (unknown Variable). The roots of a quadratic equation are the values of the variables like x, y, z etc. that satisfy the equation. A Hindu mathematician named Brahmagupta developed the general solution to the quadratic equation around the year 700 AD. He employed numbers that were irrational and recognized two roots in the solution, among other factors.

In every field where calculate the speed, area, or profit is necessary, quadratic equations must be applied directly or indirectly. Before beginning a job, construction personnel employ quadratic equations to determine the area. In addition, they are referred to as the "solutions" or "zeroes" of the quadratic equation.

Let us consider the quadratic equation ax2+ bx + c = 0 where a is not equal to zero.

If we assume that α and β are the roots of the equation ax2+ bx + c = 0. Here roots can be found as”:

• Sum of the Roots:
• Product of the Roots:
• Difference of the Roots:

### Sum of the Roots:

The sum of the roots can be defined as, α + β = -b / a where b is the co-efficient of x and a is the co-efficient of x2

### Product of the Roots:

The product of the roots can be defined as, α x β = c / a where c is the co-efficient of constant term and a is the co-efficient of x2

### Difference of the Roots:

The difference of the roots can be defined as,

(α – β) ^2 = (α +β) ^2 – 4αβ = b^2 /a^2 – 4c / a = (b2-4ac)/ a^2 = ((√b2-4ac)/a=√D/a where D is discriminant.

### Important Note:

The quadratic equation can be expressed in term of sum and product of roots. The second term coefficient negative divided by the leading coefficient essentially equals to the sum of roots of quadratic equation.

## Methods of Finding the Roots of Quadratic Equations:

Different techniques can be used to find the roots of quadratic equations some discussed here like:

• Factorizing
• Complete the Square

### Factorizing:

Factorization is the first and most basic technique for solving quadratic problems. Factoring can be used to simplify some algebraic expressions. Therefore, we may reduce the value of x from these equations. Expanding brackets in reverse is a procedure known as factorizing.

Completely factorizing an equation uses what is known it in brackets by eliminating the factors with the highest common denominator. Finding the highest common factor between all of the terms in the expression is the simplest technique to factories.

Example:

To solve the quadratic equation x2 + 6x + 9 = 0 by factorization.

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Steps:

1. Choose the two digits such that if we multiplied these two digits it gives 9x2 and if we add them it gives plus 6x.
2. Then take the common factor between them.
3. We obtain two factors these two factors are kept equal to zero.
4. By this we can obtain the values of unknown variables.

Solution:

Given that:

• x2 + 6x + 9 = 0

x2+3x+3x+9 = 0

• By taking the common:

x (x + 3) + 3 (x +3)

(x + 3) + (x + 3)

(x + 3) = 0 and (x + 3) = 0

It gives,

x = -3 and x = -3

The solution set is given as,

Solution Set is {-3, -3}.

### Complete the Square:

Steps to Solve the Quadratic Equation by complete the square:

• Arrange the given equation in the form x2+b = c
• Add the term required to complete the square to both sides of the equation.
• Compute the multiplicand of the perfect square.
• Apply the square root property to the resulting equation to solve it.

Example:

Solve the given equation x2 + 6x + 9 = 0 by complete square method.

Solution:

• Rearrange the equation x2 + 6x = -9
• Add 2 on both sides of the equation. x2 + 6x + 2 = -9 + 2
• x2 + 6x + 2 is the complete square of (x + 2)2

So, (x + 2)2= -7

• By taking the square root on both sides of equation √-7= √(x + 2)2
• It gives x + 2 = 7iota because square root of -1 is complex which can be written in the form of iota.
• Now find the value of x by x = 7iota – 2

• Arrange the given equation in the form ax2+b + c = 0
• Apply the quadratic formula (-b ± √ b2 - 4ac) / 2a

Example:

Solve the quadratic equation x2 + 6x + 9 = 0 by quadratic formula

Solution:

• x2 + 6x + 9 = 0
• Where a = 1 , b = 6 and c = 0
• Apply the quadratic formula (-b ± √ b2 - 4ac) / 2a

(-6 ± √36 – 0 / 2

-6 ± 6 / 2

-6 + 6 / 2 and -6 – 6 /2

0 and -6

So, the solution set is {0, -6}