Identifying Quadratic Expressions

Definition of Quadratic Expressions

A quadratic expression is an algebraic expression having a degree of 2.

Identifying One Variable Quadratic Expressions

An algebraic expression in one variable is quadratic when:
  • The variable has an exponent of 2 on it
  • The variable does not have an exponent greater than 2
  • The variable does not have a negative exponent
In simple words the above three statements mean:
  • There is at least one term in the expression with `x^2` in it
  • There is no term in the expression with `x^3` or higher exponents in it
  • The expression does not have `x` in the denominator

Example

The algebraic expression `x^2 + 2x + 3` is quadratic because
  • 'x' has an exponent of '2' on it
  • 'x' does not have an exponent greater than 2 on it, and
  • 'x' has an exponent of 1 in the term '2x' and an exponent of 0 in the term '3'

Counter Examples

  • The expression `x^3 + 3x^2` is not quadratic because 'x' has an exponent greater than 2 in it.
  • The expression `x^2 + 1/x` is not quadratic because 'x' is in the denominator (and hence has negative exponents)
  • The expression `0x^2 + 3x + 2` is not quadratic because there is no term with `x^2` in it (the term `0x^2` equals 0 so it is not considered)

Identifying Two Variable Quadratic Expressions

A quadratic expression can have two variables, given that 
  • the sum of exponents of both variables in any term is never greater than 2, and
  • there is at least one term whose degree is 2
  • no variable is present with a negative exponent

Example

`x^2 + xy + y^2` is a quadratic expression because
  • `x` and `y` do not have exponents greater than 2 in any term
  • sum of exponents of x and y in the term `xy` is 2

Counter Examples

  • `x^2y + xy + 1` is not a quadratic expression because the sum of exponents of `x` and `y` in the term `x^2y` is 3, which is greater than 2
  • `x/y + x^2` is not a quadratic expression because `y` is in the denominator, thus making it a rational expression

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