How to Solve Quadratic EquationsHow to Solve Quadratic Equations
is a polynomial equation of second degree. The general form of a
quadratic equations 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
Discriminant of quadratic equation
The following table shows the nature of the roots of a How to Solve Quadratic Equations
Discriminant [b] 2 [ -4ac] Square root
= 0 perfect
rational and equal
perfect or not perfect
rational and un equal (or)
irrational and unequal roots
< 0 not perfect
complex and conjugate roots in pair
Formation of a quadratic equation
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]
Maximum and minimum values of a Quadratic expression
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 = - 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 = - b/2a.
Quadratic Equation Formula
The general form of a quadratic equations is ax2+bx+c = 0 .
The set of al solutions of a quadratic equation is cal ed 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+-sqrt(b] 2 [-4ac))/(2a)]
[where b] 2 [ -4ac] [ is called the How to Solve Quadratic Equations
.] [ A quadratic equation has two
Solving quadratic equations
Here is the examples on Solving a quadratic equation based on methods to solve it-
Solve x2 + 2x = 15 by factoring.
Rewrite equation in standard form: x 2 + 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
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
3x ?1+1= 3 +1 or 3x -1+1= -3 +1