Subtract. State the difference in simplest form.
$\frac{x^2}{x-6}-\frac{36}{x-6} \quad x \neq 6$
A. $x-6$
B. $\frac{x+6}{2}$
C. $x+6$
D. $\frac{x-6}{2}$
Answer (2)
Combine the fractions: x − 6 x 2 − 36 .
Factor the numerator: x − 6 ( x − 6 ) ( x + 6 ) .
Cancel the common factor: x + 6 .
The simplified expression is x + 6 .
Explanation
Understanding the Problem We are asked to subtract two rational expressions with the same denominator. The expressions are x − 6 x 2 and x − 6 36 , with the condition x = 6 . We need to simplify the result.
Combining Fractions To subtract the two rational expressions, we combine them into a single fraction since they have the same denominator: x − 6 x 2 − x − 6 36 = x − 6 x 2 − 36
Factoring the Numerator Now, we factor the numerator, which is a difference of squares: x 2 − 36 = ( x − 6 ) ( x + 6 ) So our expression becomes: x − 6 ( x − 6 ) ( x + 6 )
Simplifying the Fraction We can now simplify the fraction by canceling the common factor ( x − 6 ) in the numerator and the denominator, keeping in mind that x = 6 : x − 6 ( x − 6 ) ( x + 6 ) = x + 6
Final Answer Therefore, the simplified expression is x + 6 .
Examples
Understanding how to simplify rational expressions is useful in many areas, such as physics and engineering, where complex formulas can be simplified to make calculations easier. For example, when analyzing the motion of objects or designing electrical circuits, simplifying rational expressions can help in finding solutions more efficiently. This skill is also crucial in calculus when dealing with limits and derivatives of rational functions. By simplifying these expressions, we can more easily understand and manipulate them, leading to more efficient problem-solving.
The difference of the two fractions simplifies to x + 6 . The correct answer is option C. This is found by factoring and canceling common terms in the expression.
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