subject: Quadratic Formula Problem [print this page] Introduction to quadratic formula problem:
Definition:
An equation in which the highest power of the variable is 2 is called a quadratic equation.
ax^2+bx+c=0, where a, b, c are constants, is a general quadratic equation.
Any grouping of constants and variables generated by applying finite number of the elementary operations-addition, subtraction, multiplication, division, or the extraction of roots -is called an algebraic expression. For example,
$frac{3x^{4}+5x^{2}-sqrt{2x+4}}{9}$, $xy^{2}+5x^{2}z-sqrt[3]{z}$, etc.
are algebraic expressions. Thus, ax^2+bx+c is also an algebraic expression.
Quadratic expression in One variable(or unknown)
A polynomial of the form ax^2+bx+c(a'!=' 0) is called a quadratic expression in the variable x. This is a polynomial of the second degree.
For example, (i) 5x^2-7x+2 and (ii)2x^2+1 are quadratic expressions in x.
But $frac{6}{x^{2}}-frac{1}{x}+7$ is a quadratic expression in 1/x, and 9x-2 is not a quadratic expression.
In the quadratic expression ax^2+bx+c, a is co-efficient of x^2, b is the co-efficient of x and c is the constant term.
Classification of Quadratic equations:
Quadratic equations are classified into two categories:
(i) Pure quadratic equations(of the form ax^2+c=0, i.e., b=0 in ax^2+bx+c=0)
(ii) Affected (or affected) quadratic equations(of the form ax^2+bx+c=0, b'!=' 0).
For example, (i)x^2-4=0 and (ii)3x^2-1=0 are pure quadratic equations, while (iii) x^2-2x-8=0 and (iv) 3x^2+x=0 are affected quadratic equations.
Quadratic Formula for the Std Equation
Solution of the equation ax^2+bx+c=0
Proof:
ax^2+bx+c=0 (a'!=' 0)
ax^2+bx=-c (transposing the constant term)
x^2+'(b)/(a)' x= '(-c)/(a)' (dividing by the co-efficients of x^2)
