**Quadratic Equation**

Contents

__Quadratic Equation__

A quadratic equation in the variable x is an equation of the form ax2 + bx + c = 0, where a, b, c are real numbers, a ≠ 0. E.g.: 2x2 – 3x + 7 = 0,

Application:

- Used to find effective resistance of a circuit
- Used in the field of communications
- Used to find the field of architecture
- Used in the field of finance to find demand supply relation
- Used to find the projectile of ball throw or bomb throw
- Used to find speed of train, boat etc

It is believed that Babylonians were the first to solve quadratic equations. Greek mathematician Euclid developed a geometrical approach for finding solutions of quadratic equations.

Solving of quadratic equations, in general form, is often credited to ancient Indian mathematicians.

Any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2, is a quadratic equation.

But when we write the terms of p(x) in descending order of their degrees, then we get the standard form of the equation.

That is, ax2 + bx + c = 0, a ≠ 0 is called the standard form of a quadratic equation.

__Convert Statement to Quadratic Equation__

John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.

Solution: Let john has x marbles, then Jivanti has 45-x marbles, since total number of marbles is 45.

Now both of them lost 5 marbles each, thus now marble count is

- John : x-5
- Jivanti : 45-x -5 or 40-x

Given that product of new marble count is 124, that is Marbles with john * Marbles with Jivanti = 124.

(x-5) * (40-x) = 124

Or x2 – 45x + 324 = 0 , where x is the count of marbles with John.

**Numerical 1**: Check if x(x + 1) + 9 = (x + 2) (x – 2) is a quadratic equation

Solution: On expanding the equation, we get

x2 + x + 9 = x2 + 2x -2x -4

x + 13 = 0.

Since, it is not of the form ax2 + bx + c = 0, it is not a quadratic equation.

__Solution of a Quadratic Equation by Factorization__

For a quadratic equation ax2 + bx + c = 0, a ≠ 0

if aα2 + bα + c = 0. We also say that

**x =**α**is a solution of the quadratic equation**if aβ2 + bβ + c = 0. We also say that

**x =****α****is a solution of the quadratic equation.****Thus if**α & β are solution of the quadratic equation ax2 + bx + c = 0. Then we can write

ax2 + bx + c = 0 = (x- α) (x-β), where α & β are solution of this quadratic equation.

In case of factorization method, we re-write the quadratic equation in the form (x- α) (x-β).

Example: Find solution for quadratic equation x2 – 5x + 6 =0

By splitting the middle term we can rewrite the equation as

x2 – 3x -2x + 6 =0 as (-3x * -2x = x2 * 6)

Or x(x-3) -2 (x-3) 0

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

Or x=3 or x= 2 are solutions for this quadratic equation.

__Solution of a Quadratic Equation by Completing the Square__

In this case, we try to reduce a quadratic equation in the form (px+q)2 = n2

This x = (+n –q)/p or (-n –q)/p

For Example, lets solve quadratic equation x2 + 4x – 5=0

We can rewrite this equation as

(x2 + 4x + 4) -4 – 5=0

or (x+2)2 = 32

or X = +3 -2 or -3-2

or x =1 or -5

__Nature of Roots__

A quadratic equation ax2 + bx + c = 0 has

(i) two distinct real roots, if b2 – 4ac > 0,

(ii) two equal real roots, if b2 – 4ac = 0,

(iii) no real roots, if b2 – 4ac < 0.

Since *b*2 – 4*ac *determines whether the quadratic equation *ax*2 + *bx *+ *c *= 0 has real roots or not, *b*2 – 4*ac *is called the **discriminant **of this quadratic equation.

**Also Read :**

### MATHS Revision Notes

Chapter:01 Real Numbers System

Chapter:02 Polynomials

Chapter:03 Pair of Linear Equations in Two Variables

Chapter:04 Quadratic Equation

Chapter:05 Arithemetic Progressions

Chapter:06 Triangles

Chapter:07 Coordinate Geometry

Chapter:08 Introduction to Trignometry

Chapter:09 Some Application Of Trignometry

Chapter:10 Circles

Chapter:11 Constructions

Chapter:12 Area Related to Cirles

Chapter:13 Surface Area Volume

Chapter:14 Stastistics

Chapter:15 Probability