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## Introduction

The polynomial equation whose highest degree is two is called a quadratic equation.

General form of quadratic polynomial is ax2 + bx + c, a ≠ 0. When we equate this polynomial to zero, we get a quadratic equation.

Thus, quadratic equation is ax2 + bx + c= 0 where a, b and c being constants (a ≠ 0).

Example:

x2 + 4x + 4 = 0 (standard form of quadratic equation)

2x2 – 64= 0 (Missing linear coefficient)

x2 -7x =0  (Missing constant term)

(x + 2)(x - 3) = 0 (Quadratic equation example in factor form)

## Quadratic equation have two real Solutions.

An interesting fact about quadratic equations is that it can have up to two real solutions. Solutions are values of variable where the quadratic equals 0. Real solutions mean that solutions are not imaginary and they are real numbers.

The solutions of a quadratic equation ax2 + bx + c = 0 Where a, b and c being constants (a ≠ 0) are given by the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

Here (b² - 4ac) is called discriminant of quadratic equation. That is denoted by D.

D= (b² - 4ac)

Discriminant is used to determine the nature of roots of the quadratic equation.

### Nature of roots of the quadratic equation

·         If D > 0, then the equation ax2 + bx + c = 0 has two real and distinct solutions (roots).

·         If D = 0, then the equation ax2 + bx + c = 0 has only one real solution (root)or equal roots.

·         If D < 0, then the equation ax2 + bx + c = 0 has two distinct complex roots.

By using the discriminant, we can determine the number of solutions of quadratic equations without actually solving it.

### Sum and product of roots

For any quadratic equation ax2 + bx + c = 0, where a, b and c being constants (a ≠ 0).

·         The sum of roots = -b/a

·         The product of the roots = c/a.

## Graph of the quadratic equation

·         The graph of a quadratic function is a U-shaped curve termed a parabola.

·         The sign on the coefficient of “a” the quadratic function dictates whether the graph opens up or down.

·          If a<0, the graph opens down and if a>0 then the graph opens up.

### Method 1: Factoring method

To solve a quadratic equation by factoring method,

1.    Put all terms on LHS side of the equal sign zero on the RHS side.

2.    Factorize LHS side polynomial.

3.    Set each factor equal to zero.

4.    Find value of variable by equating each factor with zero.

5.    Check by inserting your solutions in the original equation.

Example:  Solve x 2 – 6 x = 16.

Step1: Put all terms on LHS side of the equal sign zero on the RHS side

x 2 – 6 x = 16

=> x 2 – 6 x – 16 = 0

Step 2: Factorize LHS side polynomial.

=> ( x – 8)( x + 2) = 0

Step 3: Set each factor equal to zero.

x – 8)=0

=> x=8

x + 2) =0

=> x=-2

Step5: Check by inserting your solutions in the original equation.

When x=8, LHS=(8)2- 6*8 =64-48=16 Here ,LHS =RHS

When x=-2, LHS=(-2)2- 6*(-2) =4+12=16 Here ,LHS =RHS

Thus, roots of   x 2 – 6 x = 16 are 8,-2.

### Method 2: Using standard formula for roots of quadratic equation

Many quadratic equations cannot be solved by factoring method. This is required when the roots are not rational numbers.

A second method of solving quadratic equations needs the use of the following formula:

a, b, and c are taken from the quadratic equation by comparing with general form of quadratic equation (ax 2 + bx + c = 0)

Example:  Solve x 2 – 6 x = 16.

Given quadratic equation is x 2 – 6 x = 16 or x 2 – 6 x – 16=0

Now comparing with standard form ax 2 + bx + c = 0 we get

a=1, b=(-6) and c=(-16)

using formula for x,

X=[ -(-6) ± √({-6}² - 4*1*{-16}]/2*1

X=(6±10)/2

When we take x=(6+10)/2=>x=8

When we take x=(6-10)/2=>x=-2