Quadratic Equations constitute an important part of Algebra. This portion lays the foundation of various important topics

Topics Include from this Section are :

Discriminant of a Quadratic Equation
Polynomial Equation of Degree n
The Method of Intervals Wavy Curve Method
Interval in which the Roots Lie
Maximum And Minimum Value of a Quadratic Expression
Resolution of a Quadratic Function Into Linear Factors

the Q Quadratic Equations Solutions are given below


It is an equation of the form ax² + bx + c =0, where a , b, and c are real numbers and a is not equal to 0. If a=0, the equation will reduce to bx +c =0, which is a linear equation, and not a quadratic equation.

solution

ax² + bx + c =0

⇒ a( x² + (b/a)x + c/a) =0

⇒( x² + (b/a)x + c/a) =0 , as a is not 0.

⇒ (x² + 2 *( b/2a) *x + ( b/2a)² - ( b/2a)² + c/a ) = 0 -1


As of now , we know the solution of the form (x+A)²=B². If we carefully observe the above equation,we find that both x² and x are present. So we must put this in the form(x+A)².
(x+A)²=x² + 2Ax + A². The coefficient of x is b/a , which must be equal to 2A. Thus, A=b/2a.

Now eq 1 can be written as
( x + b/2a)² - ( b² -4ac)/4a² =0

⇒ ( x + b/2a)² = ( b² -4ac)/4a²

⇒ x + b/2a = ± √( ( b² -4ac)/4a²)

⇒ x + b/2a = ± √( b² -4ac) /2a

⇒ x = ( -b ± √( b² -4ac) ) /2a

( b² -4ac ) is denoted by D.

Thus we get two solutions- x = (-b + √D) /2a, and
x = (-b - √D) /2a