Divide and simplify.
$\frac{x^2-12 x+36}{x^2-x-30} \div \frac{x^2-36}{9}$
Answer (2)
Rewrite the division as multiplication by the reciprocal.
Factor the quadratic expressions.
Substitute the factored expressions into the expression.
Cancel common factors to get the simplified expression: ( x + 5 ) ( x + 6 ) 9 .
Explanation
Understanding the Problem We are asked to divide and simplify the expression x 2 − x − 30 x 2 − 12 x + 36 ÷ 9 x 2 − 36 . This involves dividing rational expressions, which can be simplified by factoring and canceling common factors.
Rewrite as Multiplication First, we rewrite the division as multiplication by the reciprocal: x 2 − x − 30 x 2 − 12 x + 36 ÷ 9 x 2 − 36 = x 2 − x − 30 x 2 − 12 x + 36 ⋅ x 2 − 36 9
Factor Quadratic Expressions Next, we factor the quadratic expressions: x 2 − 12 x + 36 x 2 − x − 30 x 2 − 36 = ( x − 6 ) ( x − 6 ) = ( x − 6 ) 2 = ( x − 6 ) ( x + 5 ) = ( x − 6 ) ( x + 6 )
Substitute Factored Expressions Now, we substitute the factored expressions into the expression: ( x − 6 ) ( x + 5 ) ( x − 6 ) 2 ⋅ ( x − 6 ) ( x + 6 ) 9
Cancel Common Factors We cancel common factors: ( x − 6 ) ( x + 5 ) ( x − 6 ) ( x − 6 ) ⋅ ( x − 6 ) ( x + 6 ) 9 = x + 5 x − 6 ⋅ ( x − 6 ) ( x + 6 ) 9 = ( x + 5 ) ( x + 6 ) 9
Final Answer Thus, the simplified expression is ( x + 5 ) ( x + 6 ) 9 .
Examples
Rational expressions are used in many areas of science and engineering to model relationships between different quantities. For example, in physics, they can be used to describe the motion of objects or the behavior of electrical circuits. In economics, they can be used to model supply and demand curves. Simplifying rational expressions makes it easier to analyze these relationships and make predictions about the system being modeled. For instance, if you're designing a bridge, you might use rational expressions to model the forces acting on the bridge and ensure it can withstand the load.
The expression simplifies to ( x + 5 ) ( x + 6 ) 9 after rewriting the division as multiplication and factoring the quadratic expressions. Common factors are then canceled to reach the final simplified form.
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