Definition of quadratic expressions

General definition
"An algebraic expression in which the highest exponent is 2 is referred to as a Quadratic Expression"
The above definition implies that in a Quadratic Expression, the exponent on the variable can neither be greater than 2 nor smaller than 2.

For example, the following algebraic expressions are quadratic expressions because their highest exponent is exactly equal to 2:
  • x2 + 3x + 5
  • 1 + x2
  • y2 - 5xy
Mathematical definition
"An algebraic expression of the form ax2 + bx + c, in which a, b and c are integers and a is not equal to zero is called a Quadratic Expression" 
The above definition is a more accurate definition of quadratic expressions. You can compare all quadratic expressions to ax2 + bx + c and get the values of a, b and c. For example, as follows:
  • 2x2 + 3x + 2; a = 2, b = 3 and c = 2
  • x2 + 1; a = 1, b = 0 and c = 1
  • 3y2 - 2y; a = 3, b = -2 and c = 0
  • 4x2; a =  4, b = 0 and c = 0
In the above equations, the values of a are called coefficients of x squared; the values of b are called coefficients of x, and the values of c are called the constant terms of the respected Quadratic Expression.

You must have noticed these three things in the above examples:
  1. The value of a is never equal to zero
  2. The values of b and c can be zero
  3. The values of a, b and c can be negative  
Common Questions 

  • Why can't the value of 'a' (that is, coefficient of x2) be zero?
... because if the value of 'a' will be zero, then the value of 'ax2' will also be zero, and hence the expression will not remain Quadratic. 
  • Why can the values of 'b' and 'c' be equal to zero?
... because the degree, or the highest exponent, of the Quadratic Expression does not change if the values of 'b' and/or 'c' are equal to zero.

Note: A quadratic expression is also called as a second degree expression because the degree, or highest exponent, of a quadratic expression is always two.

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