** Quadratic equation** is the second-degree equation in which one variable contains the variable with an exponent of 2.

Its general form is **ax**^{2}** + bx + c = 0, a ≠ 0**

## Types of Quadratic Equations

There are following two special types of quadratic equations: –

- Quadratic Equations with no term in x
- Quadratic Equations with no constant term

See also: Types of Equations

### 1. **Quadratic Equations with no term in x:**

If there is a quadratic equation with no term in x then we can move the constant to the other side. For example,

**Example:**

Solve
x^{2} – 2 = 0

**Solution:**

x^{2} – 2 = 0

x^{2} = 2

x = 2 or x = –2

### 2. **Quadratic Equations with no constant term:**

**Example:**

Solve
x^{2} – 16x
= 0

**Solution:**

x^{2} – 16x
= 0

By taking common x

x(x – 16) = 0

As the product of the two factors is 0, therefore, one or both of the factors is zero

x(x) – 16 = 0. Hence, x = 0 or x – 16 = 0

Therefore, the two solutions are

x = 0, or x = 16

You may also read: Linear Equation

**Methods of Solving Quadratic Equations**

Quadratic equations can be solved by using the following four methods:

- By factorization
- By completing the square
- By using quadratic formula
- By using graphs

But, here we discussed the first three methods of solving the quadratic equations. However, the solution of the quadratic equation by factorization and use of quadratic formula are particularly significant.

**a) Solution of Quadratic Equations by Factorization**

The procedure of solving any Quadratic Equation by factorization is same as used in above example that if we find 0 as the product of two numbers then at least one of the numbers must be 0.

This means that, if AB = 0 then A = 0 or B = 0

In Factorization, first we learn, how to factorize monic quadratic equation then we learn how to factorize a non-monic quadratic equation.

A non-monic quadratic equation is an equation of the form x^{2 }+ bx + c. we always try to find out two numbers whose sum is ‘b’ and whose product is ‘c’. Now we apply this method to solve some monic quadratic equations.

**Example
No.1:**

Solve
x^{2} – 6x – 16 = 0

**Solution:**

First, we try to find out two numbers whose product is 16 and whose sum is –6. obviously, –8, + 2 are our required numbers. So, we can factor as:

x^{2}
– 6x + 16 = 0

x^{2} – 8x + 2x – 16 = 0

x(x – 8) + 2(x – 8) = 0

(x – 8)(x + 2) = 0

As the product of the two factors is 0, therefore, one or both of the factors is zero

x – 8 = 0 or x + 2 = 0

Therefore,

x = 8 or x = 2

**Example
No.2:**

Solve
x^{2} – 7x + 12 = 0

**Solution:**

First, we try to find out two numbers whose product is 12 and whose sum is –7. obviously, –4, –3 are our required numbers. So, we can factor as:

x^{2}
– 7x + 12 = 0

x^{2} – 4x – 3x +12 = 0

x(x – 4) – 3(x – 4) = 0

(x – 4)(x – 3) = 0

As the product of the two factors is 0, therefore, one or both of the factors is zero

x – 4 = 0 or x – 3 = 0

Therefore,

x = 4 or x = 3

A non-monic quadratic equation is an equation of the form ax^{2 }+ bx + c. we utilize the same method to solve a non-monic quadratic equation. Now we apply this method to solve few non-monic quadratic equations.

**Example
No.1:**

Solve
2x^{2} – 5x – 12 = 0

**Solution:**

Here, we multiple 2 and 12 to give 24 and find the two numbers to give 24 and add to give –5 numbers. Obviously, –8, +3 are our required numbers. So, we can factor as:

2x^{2} – 5x – 12 = 0

2x^{2} – 8x + 3x – 12 = 0

2x(x – 4) + 3(x – 4) = 0

(x – 4)(2x + 3) = 0

As the product of the two factors is 0, therefore, one or both of the factors is zero

x – 4 = 0 or 2x + 3 = 0

Therefore,

x = 4 or x = –3/2

**Example
No.2:**

Solve
3x^{2} + 25x – 18 = 0

**Solution:**

Here, we multiple 3 and 18 to give 54 and find the two numbers to give 54 and add to give +25 numbers. Obviously, +27, –2 are our required numbers. So, we can factor as:

3x^{2} + 25x – 18 = 0

3x^{2} + 27x – 2x – 18 = 0

3x(x + 9) – 2(x + 9) = 0

(x + 9)(3x – 2) = 0

As the product of the two factors is 0, therefore, one or both of the factors is zero

x + 9 = 0 or 3x – 2 = 0

Therefore,

x = –9 or x = 2/3

**b) Solution of Quadratic Equations by completing the square**

Now we are familiar with quadratic equations having one or two solutions that were rational but there are several quadratic questions that have irrational solutions or in few cases, no real solutions. Therefore, in order to handle the more general quadratic questions, we employ a most commonly method of solution, which is known as completing the square. This method can also be utilized for finding the radius and centre of a circle, minimum or maximum of a quadratic function, putting integrals into typical form in Calculus, etc.

**Example
No.1:**

Solve
x^{2} +
2x − 10 = 0

**Solution:**

First, we focus on x^{2} + 2x to find a number
that must be added to *x*^{2} + 2*x *to make the expression
into a perfect square. Take half of the coefficient and square it in both cases
i.e. x positive or x negative, here it is 1, therefore, x^{2} + 2x + 1 = (x + 1)^{2}. Hence,

x^{2} +
2x − 10 = (x^{2} + 2x + 1) – 1 – 10 (add and subtract 1)

= (x + 1)^{2} – 11

(x + 1)^{2} = 11

Now take the positive and negative square roots to obtain

x + 1 = √(11 ) or x + 1 = – √11

x = – 1 + √(11 ) or x= – 1 – √(11 )

**Example
No.2:**

Solve
x^{2} +
4x − 10 = 0

**Solution:**

x^{2}
+ 4x − 10 = 0

x^{2}
+ 4x + 4 – 4 − 10 = 0
(complete the square by utilizing the technique mentioned in above
example )

(x +
4)^{2} = 4 + 10

(x +
4)^{2} = 14

x + 4 = 14 or x + 4 = −14

Therefore,

x = 10 or x = −18

**c) Solution of Quadratic Equations by using Quadratic Formula**

General quadratic equation is ax^{2 }+
bx + c = 0

We utilize this formula to solve the given equation,

X = (-b±√(b^2-4ac)) / 2a

Now, we apply this formula to solve few quadratic equations.

**Example
No.1:**

Solve
x^{2} − 9x − 4 = 0

**Solution:**

x^{2} − 9x − 4 = 0

Here, a = 1, b = -9 and c = -4

Therefore,

X = (-9±√(81-4(-4))) ⁄ (2(1))

X = (-9±√(81-16)) / 2

X = (-9±√65) / 2