What is Quadratic Equation? How to solve it, explain with examples.

In mathematics, a quadratic equation is defined as an equation with a degree of 2, which means that the highest exponent of this function or equation is 2. It is a polynomial equation in a single variable.

Standard form of Quadratic Equation:

The standard form of a quadratic equation is:

ax² + bx + c = 0

where b, and c are real numbers and a cannot be 0. Any number or constant when multiplied by 0 will also make the whole product 0. Therefore, the value of a will never be 0.The terms a, b and c are also called quadratic coefficients.

Examples of quadratic equations are:

• 2x² - 3x + 7 = 0
• -x² + 8x + 3 = 0
• ax² + 8ax + 7 = 0

Since, the quadratic equation involves only one variable hence it is also called as univariate. The power of a variable is always a non-negative integer.

How to Solve Quadratic Equations?

The solutions to a quadratic equation are the values of x that make the equation true.These are solved by determining roots of the equation. There are three main ways to solve a quadratic equation:

1. Factorization
2. Quadratic Formula
3. Completing the square

Now we are going to disucss each seprately with examples.

1- Solve Quadratic Equation by Factoring

           Factoring involves figuring out which terms should be multiplied to create a mathematical expression. The process of factoring quadratic equations can be carried out using these methods with example:

Example:

x² + 5x = - 6 

We have to convert this equation into standard form (zero on one side of the equation).

x² + 5x + 6 = 0 

Now we have to factor the non-zero side of the equation.

x² + 3x + 2x + 6 = 0 

Take common from each pair set.

 x (x+3) + 2 (x+3) = 0

 (x+3) + (x+2) = 0

After taking common we have two factors, set each factor equals to zero.

 (x+3) = 0

(x+2) = 0

Solve each factor

x = -3 ,  x = -2

The solution set is:

{-3 , -2}

2- Solve Quadratic Equation by Quadratic Formula

          To solve any equation using quadratic formula, we need to convert equation into ax² + bx + c = 0 form. Here a,b and c are coefficients of the equation.

Quadratic Formula =  -b ± √ (b² - 4ac) / 2a

Here is the step by step guide with an example,

Example

x² + 9x + 18 = 0

a = 1 ,  b = 9 , c = 18

We have to put these coefficient values in above quadratic formula:

x = -9 ± √ [9² - 4(1)(18)] / 2(1)

          x = -9 ± √ (81-72) / 2

  x = -9 ± √ (9) / 2

x = -9 ± (3) / 2

We have to solve it separately:

x = (-9 + 3) / 2

x = -6 / 2

x = -3

x = (-9 - 3) / 2

x = -12 / 2 

x = -6

The solution set is:

{-6 , -3}

3- Solve Quadratic Equation by Completing Square

              Still we have solved quadratic equations either by factorization or by quadratic formula. These methods are simple and easy to understand but are applicable to possible equations. To overcome the difficulties we have a third method named as completing the square. Let's start this method by solving an example with steps.

Example

2x² + 4x - 6 = 0

First of all we have to find the coefficient of x² in the equation. In this case 2 is the coefficient of x². We have divide whole equation with 2 (coefficient of x²).

2x² / 2 + 4x / 2 - 6 / 2 = 0 / 2

The equation after division becomes:

x² + 2x - 3 = 0

After division move constant to right side of the equation:

x² + 2x = 3

Now find the coefficient of x and divide it by 2 (or multiply with 1/2). Here, 2 is the coefficient with x.

2 * 1 / 2  = 1

Now add the square 1 to both sides of the equation.

x² + 2x + 12 = 3 + 12

On left side of the equation the expression itself becomes a square formula.

(x+1)² = 3 + 1

(x+1)² = 4

√(x+1)² = √4

By taking square roots on both sides the equations becomes: 

(x+1) = ±2

We have to solve it separately:

(x+1) = +2

x = 2 - 1

x = 1

And,

(x+1) = -2

x = -2 - 1

x = -3

The solution set is:

{-3 , 1}