In this blog, we will learn about Quadratic Equations, methods of solving a quadratic equation, and the quadratic formula with the help of solved examples. A quadratic equation is an equation that contains at least one squared variable. In our daily life, quadratic equations play a major role in calculating speed, figuring out the area, or determining profit.

**What is a Quadratic Equation?**

A quadratic equation in the variable x is an equation of the form ax^{2} + bx + c = 0, where

a, b, c are real numbers, a ≠ 0. Quadratic equation formula helps us solve quadratic equations.

Here is an **example of quadratic equation;** 5x^{2} – 3x + 3 = 0

The power 2 makes it quadratic. The name **Quadratic** comes from “quad” meaning Square, because the variable gets squared (like **x ^{2}**). But sometimes a quadratic equation doesn’t look like that!

**Standard Form of Quadratic Equation**

This is the equation in disguise form **x ^{2} = 3x – 1,**

Move all terms to left hand side,

In standard form x^{2} – 3x + 1, where a = 1, b = -3 and c = 1.

This equation is also in disguise form,

Multiply by x square,

We get the** standard form of quadratic equation, 5x ^{2} + x – 1 = 0**

Where a = 5, b = 1 and c = -1

**The standard form of a quadratic equation** is ax^{2} + bx + c = 0, where a, b, and c are known values or coefficients but a is not equal to 0 and x is unknown or the variable.

**Roots of Quadratic Equation: **Roots of the quadratic equation are the values of x satifying the equation.

**Solved Examples of Quadratic Equation**

Represent this situation mathematically:

**1. A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was Rs 750. We would like to find out the number of toys produced on that day.**

**Answer.** Let the number of toys produced on that day be x.

Therefore, the cost of production (in rupees) of each toy that day = 55 – x

So, the total cost of production (in rupees) that day = x into (55 – x)

Therefore, x into (55 – x) = 750

i.e., 55x – x^{2} = 750

which implies – x^{2} + 55x – 750 = 0

this gives x^{2} – 55x + 750 = 0

Therefore, the number of toys produced that day satisfies the quadratic equation:

**x**^{2}** – 55x + 750 = 0**

Which is the required representation of the problem mathematically

**How to solve quadratic equations? **

Learn how to solve quadratic equation by two methods here, namely, factorization method and completing the square method.

**“Solution of a Quadratic Equation by Factorisation”.**

In general, a real number α is called a root of the quadratic equation

**ax ^{2} + bx + c = 0, a ≠ 0**

If aα^{2} + bα + c = 0, we also say that x = α is a solution of the quadratic equation, or that α satisfies the quadratic equation.

**Remember: **The zeroes of the quadratic polynomial ax2 + bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same.

We know that a quadratic polynomial can have at most 2 zeros. So, any quadratic equation can have at most two roots.

Let us find the solution of a quadratic equation by factorisation, i.e., splitting the middle terms.

**Quadratic Equations Examples**

**Example 1: Factorise 3x ^{2} = 27.**

**Solution: **3x^{2} = 27

Subtracting 27 from both sides,

3x^{2} – 27 = 0

3 (x^{2} − 9) = 0

3 (x − 3) (x + 3) = 0

x − 3 = 0 or x + 3 = 0

x = 3 or x = −3

**Example 2: Factorise x**^{2}** − 5x + 6 = 0**

**Solution: **x^{2} − 5x + 6 = 0

−3×−2 = 6 – 3 + −2 = −5

So the two numbers are −3 and −2.

We use these two numbers to write −5x as −3x, – 2x and proceed to factorise as follows:

x^{2} – 5x + 6 = 0

= x^{2} – 3x – 2x + 6 = 0

= x (x – 3) – 2 (x – 3) = 0

= (x – 3) (x – 2) = 0

= x – 3 = 0 or x – 2 = 0

= x = 3 or x = 2

These are the two solutions.

**Solving Quadratic Equations by Completing the Square**

General Quadratic Equation is of the form ax square plus bx plus c is equal to 0

**Where a is coefficient of x square, b is the coefficient of x and c is the constant term.**

Let us solve it by Completing the Square.

Solving a Quadratic equation by the method of completing the square can be divided into following steps:

**Quadratic Equations Examples**

**Example: Solve the quadratic equation by completing the square.**

**2x ^{2} – 5x + 3 = 0**

**Solution:** 2x^{2} – 5x + 3 = 0

**Read More: **Triangles: Types of Triangles, Formula & Triangle Properties