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### 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