# Introduction to Quadratic equation

**What is a quadratic equation?**

*A quadratic equation in variable X is an equation in which the greatest exponent of the variable is two.*

For example:

3x^{2}+4x+2=0

**General form of quadratic equation**

ax^{2}+bx+c=0

Where a, b, c, ε R and a **≠**o or where a,b,c are real numbers and a**≠**o

If a=o

Than bx +c=0 which is not quadratic equation .it becomes linear equation.

If b=0

Then ax2+c=0 which is also a quadratic equation. But this is called a pure quadratic equation.

**Example:**

2x^{2}+5x+7=0 (general form quadratic equation)

3x^{2}+2=0 ( pure quadratic equation)

X+1=0 (linear equation)

**How to solve a quadratic equation**?

- Factorize
- Completing the square
- Using quadratic formula
- Graphing

**What is roots?**

** The solution of an equation is also called its roots.**

**Roots**

- One root will be reciprocal to the other. If a=c

The **coefficient **of x2 is equal to the constant.

For example:

2x^{2}+5x+2=0

2x^{2}+4x+x+2=0

2x(x+2) +1(x+2) =0

(2x+1)(x+2)=0

2x+1=0 x+2=0

2x=-1 x=-2

X= -1/2 x=-2

**One root is zero if c=0**

For example:

2x^{2}+4x=0

2x(x+2) =0

2x=0 x+2=0

X=0 x= -2

- The roots are equal in magnitude but opposite in direction (sign) if b=0

For example:

X^{2}-4=0

X^{2}= 4

square roots on both side

X=±2

X=2 x= -2

**If a+b+c=0**

If sum of all coefficient is equal to zero. Then 1, c/a are the roots

For example:

2x^{2}-5x+3=0

2x^{2}-2x-3x+3=0

2x(x-1)-3(x-1) =0

(2x-3)(X-1)=0

2x-3=0 x-1=0

2x=3 x=1

X= 3/2 x=1

X=1 x= 3/2

- If a-b+c =0 than the roots are -1 , -c/a

For example:

2x^{2}+5x+3=0

2x^{2}+2x+3x+3=0

2x(x+1) +3(x+1) =0

(2x+3)(x+1)=0

2x=-3 x=-1

X=-3/2 x=-1

- if one root is is p+iq than the second roots will be p-iq

- if one root is p+√q than second root p- √q

- The quadratic equation whose roots are reciprocal of the roots of ax
^{2}+bx+c=o is cx^{2}+bx+a=0

Coefficient written is reverse order

For example:

3x^{2}+7x+4=0

3x^{2}+3x+4x+4=0

3x(x+1) +4(x+1) =0

(3x+4)(x+1)=0

3x+4=0 x+1=0

3x=-4 x=-1

X= -4/3 x=-1

For example:

4x^{2}+7x+3=0

4x^{2}+4x+3x+3=0

4x(x+1) +3(x+1) =0

(4x+3)(x+1)=0

4x+3=0 x+1=0

4x=-3 x=-1

X= -3/4 x=-1

- If a=1 and b,c ε z then the roots are rational numbers and must be integers.

For example:

X^{2}+5x+6=0

X^{2}+3x+2x+6=0

X(x+3) +2(x+3) =0

(x+2)(x+3) =0

X+2=0 x+3=0

X=-2 x=-3

**ax**^{2}+bx +c=0

Sum of the roots =S= -b/a

b=coefficient of x

a=coefficient of x^{2}

Product of the roots =p=

C=constant

a=coefficient of x^{2}

## Frequently Asked Questions-FAQs

### what are Quadratic equation

*A quadratic equation in variable X is an equation in which the greatest exponent of the variable is two.*

For example:

3x^{2}+4x+2=0

### What Are the 3 Forms of a Quadratic Expression?

The three forms of quadratic expressions are

Standard form

Factored form

Vertex form

### what is the general from of quadratic equation?

The general form of a quadratic equation is ax^{2}+bx+c=0

Where a, b, c, ε R and a **≠**o or where a,b,c are real numbers and a**≠**o

### what are the method to solve a quadratic equation?

there are three basic techniques for solving a quadratic equation

by factorization

by completing square

by applying the quadratic formula

### what is Roots?

The solution of an equation is also called its roots.

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