# Quadratic equations explained in detail

Learn about quadratic equations and learn how to solve the given quadratic equations for finding the roots. There is so much to learn about Q.E. but first, we will focus on the basics.

An equation whose highest degree is 2 is called as a quadratic equation. Any quadratic equation will be having at most two roots. We can find these two roots by two methods.

## The General form of quadratic equations:

Generally, any quadratic equation will be in the form of a${ x }^{ 2 }$ + bx + c = 0 where a, b and c are real numbers or complex numbers and “a” should not be equal to zero (a ≠ 0). If suppose a = 1 then the quadratic equation is called Monic quadratic equation. In terms of “y” also we can write a general form of Q.E.

Examples:

2${ x }^{ 2 }$ – 5x + 4 = 0

9${ x }^{ 2 }$ + 3x – 12 = 0

${ x }^{ 2 }$ + 17x – 78 =0

23${ x }^{ 2 }$ – 56 = 0

16${ y }^{ 2 }$ – 4y – 65 = 0

42${ y }^{ 2 }$ + 18y – 56 = 0

${ y }^{ 2 }$ + 16y + 98 = 0

In the above examples ${ x }^{ 2 }$ + 17x – 78 =0 and ${ y }^{ 2 }$ + 16y + 98 = 0 are Monic quadratic equations.

Note:

Q.E does not have more than 2 roots.

### The Methods for finding the Roots of quadratic equations:

For any Given Q.E, roots can be find out by using below given two methods.

Factorization method

♦ Formula method

So any one method from the above two methods we can use and find the roots but that depends upon the quadratic equation. If the Q.E. has exact roots then we can use the factorization method otherwise that means if there are no exact factors then we need to use formula method. Below you can find the formula which is to be used for finding roots.

Quadratic equation formula:

The formula for finding roots of Q.E. a${ x }^{ 2 }$ + bx + c = 0 is $\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }$

Graph of the quadratic equation:

If we draw the graph for any Q.E. the shape will be a “Parabola” that means we will get “U” shape.

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