Simplify
$\sqrt{x^2-18 x}$
Answer (2)
The expression x 2 − 18 x can be simplified to x ( x − 18 ) . The domain of this expression is valid for x ≤ 0 or x ≥ 18 .
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Factor the expression inside the square root: x 2 − 18 x = x ( x − 18 ) .
Determine the domain by solving x ( x − 18 ) ≥ 0 , which gives x ≤ 0 or x ≥ 18 .
State the simplified expression with its domain.
The simplified expression is x ( x − 18 ) for x ≤ 0 or x ≥ 18 .
Explanation
Understanding the Problem We are asked to simplify the expression x 2 − 18 x . This involves finding a way to rewrite the expression in a more compact or understandable form, if possible. The expression contains a square root, which means we need to consider the values of x for which the expression is real-valued. Specifically, the term inside the square root, x 2 − 18 x , must be greater than or equal to zero.
Finding the Domain To determine the domain of x for which the expression is real, we need to solve the inequality x 2 − 18 x ≥ 0 . We can factor the quadratic expression as x ( x − 18 ) ≥ 0 . This inequality holds when both factors are non-negative or both factors are non-positive.
Analyzing the Cases Case 1: Both factors are non-negative: x ≥ 0 and x − 18 ≥ 0 , which means x ≥ 0 and x ≥ 18 . The intersection of these two inequalities is x ≥ 18 .
Case 2: Both factors are non-positive: x ≤ 0 and x − 18 ≤ 0 , which means x ≤ 0 and x ≤ 18 . The intersection of these two inequalities is x ≤ 0 .
Therefore, the domain of x for which the expression is real is x ≤ 0 or x ≥ 18 .
Simplified Expression The expression x 2 − 18 x can be written as x ( x − 18 ) . Since we are asked to simplify the expression, and we have already determined the domain, we can state the simplified expression along with its domain. There isn't a further simplification that can be done without introducing complex numbers or absolute values.
Final Answer Thus, the simplified expression is x ( x − 18 ) for x ≤ 0 or x ≥ 18 .
Examples
Consider a scenario where the area of a rectangle is given by A = x 2 − 18 x , where x is a variable related to the dimensions of the rectangle. We need to find the possible values of x for which the area is a real number. This problem is directly applicable in geometry and physics, where areas and distances must be real. Understanding the domain of such expressions ensures that the calculated areas or distances are physically meaningful.
