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 x2 in it
- There is no term in the expression with x3 or higher exponents in it
- The expression does not have x in the denominator
Example
The algebraic expression x2+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 x3+3x2 is not quadratic because 'x' has an exponent greater than 2 in it.
- The expression x2+1x is not quadratic because 'x' is in the denominator (and hence has negative exponents)
- The expression 0x2+3x+2 is not quadratic because there is no term with x2 in it (the term 0x2 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
x2+xy+y2 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
- x2y+xy+1 is not a quadratic expression because the sum of exponents of x and y in the term x2y is 3, which is greater than 2
- xy+x2 is not a quadratic expression because y is in the denominator, thus making it a rational expression
What formula would I use for x^4-x^3+27x-27
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