Quadratic Expression

An expression of the form ax2 + bx + c is called a quadratic expression. The value of this expression is usually plotted on the Cartesian plane along y axis. Then the expression is denoted as y = ax2 + bx + c. Let us find when the expression is positive, negative and zero.

As the expression is a combination of a variable and three constants and is a simple addition of three terms and does not take the form 0/0, 00, ∞/∞ , ∞−∞ , 0×∞ , 1 and ∞0 for any value of x, the function y = ax2 + bx + c is continuous and takes every value in the range −∞ to +∞ for −∞<x<+∞.

By the factor theorem we can express the expression y = ax2 + bx + c as the product of two terms because if the value of y is 0 for some value α then (x − α) is a factor the the expression. On finding one such factor the other factor can be found vary easily.

Suppose we have found two factors (x − α) and (x − β) such that α<β. Then y = a(x − α)(x − β). The expression is positive, negative or zero according to the expression containing factors if a>0.

  • 0: The expression is zero if one or all of the factors is zero i.e. x = α or x = β.
  • <: The expression is less than zero if the product of both the factors is less than zero i.e. one is positive and the other is negative. This happens when x lies between α and β.
  • >: The expression is positive if both the factors are negative or both are positive. this happens if the value of x is less than α or greater than β.
The expression reverses sign in a<0.