Quadratic Equation

Quadratic Equation is a polynomial equation of second degree. The general form of a quadratic equations is ax²+bx+c = 0.

The contributions of the ancient Indian Mathematicians to quadratic equations are quite significant and extensive. Before 800BC Indian Mathematicians constructed 'altars' based on the solutions of quadratic equation ax²+bx+c =0, Aryabhatta gave a rule to sum the geometric series which involves the solution of a quadratic equation.

The following table shows the nature of the roots of a quadratic equation with rational coefficients.

Discriminant `b`2 ` - 4ac`	Square	root
= 0	perfect	rational and equal
> 0	perfect or not perfect	rational and unequal (or) irrational and unequal roots
< 0	not perfect	complex and conjugate roots in pair

Let us form the quadratic equation whose roots are `alpha` and `beta` .

Then x = `alpha` , y =`beta` are the roots

Therefore x - `alpha` = 0 and

, y -`beta=0`

so (x - `alpha`) (y -`beta``)= 0`

An expression of the type ax²+bx+c is called " quadratic expression".

The quadratic expression ax²+bx+c takes different values as x takes different values.

As x varies from -`prop` to + `prop`ax²+bx+c

1. has a minimum value whenever a> 0.

The minimum value of the quadratic expression is (4ac-b² )/4a and it occurs at x = $\frac{-b}{2a}$.

2. has a maximum value whenever a< 0.

The maximum value of the quadratic expression is (4ac-b² )/4a and it occurs at x = $\frac{-b}{2a}$.

The general form of a quadratic equations is ax²+bx+c = 0 .

The set of all solutions of a quadratic equation is called its solution set. The values of x that make a quadratic equation true is called its roots or zeros or solutions. Quadratic equations can be solved by factorization method or by using quadratic formula

x = (-b± √(b²-4ac))/(2a)

quadratic formula

[x = (-b+-sqrt(b] 2 [-4ac))/(2a)]

[where b] 2 [ -4ac] [ is called the discriminant of the quadratic equation.] [ A quadratic equation has two roots. ]

Here is the examples on Solving a quadratic equation based on methods to solve it-

Factoring Method

Example:1

Solve x² + 2x = 15 by factoring.

Rewrite equation in standard form: x² + 2 x - 15 = 0

Factor the left side: (x + 5) (x - 3) = 0

Apply zero-product rule: x + 5 = 0 or x - 3 = 0

Solve for x in each equation: x = -5 or x = 3

Square Root Method

Example 2:

Solve equation (3x -1)² - 9 = 0.

Apply square root method: (3 x -1) ² = 9

3 x -1 = `sqrt9` or 3x - 1 = -`sqrt9`

3 x -1 = 3 or 3 x -1 = -3

Solve equations:

3x -1+1= 3 +1 or 3x -1+1= -3 +1

3 x = 4 or 3 x = -2

X = `4/3` or x = `-2/3`

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