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Chapter 1: Problem 170

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \((2 x-3)^{2}=25\) is equivalent to \(2 x-3=5\)

Short Answer

Expert verified
The statement is false. The correct statement should be that the equation \((2 x - 3)^2 = 25\) is equivalent to \(2 x - 3 = 卤 5\), not \(2 x - 3 = 5\). Furthermore, the solutions to the correct equation are \(x = 4\) and \(x = -1\).

Step by step solution

01

Original Statement

First, validate original equation: \((2 x - 3)^2 = 25\). The statement states this is equivalent to \(2 x - 3 = 5\). Basically, we would be saying the square root of 25 is only 5. However, square roots have two solutions, both positive and negative.
02

Solve the Correct Equation

Now, solve the correct equation for \(2 x - 3\). Square root of 25 is either 5 or -5, so the real solutions are \((2 x - 3 = 5)\) and \((2 x - 3 = -5)\). Let's solve both equations respectively. For \(2 x - 3 = 5\), \(2x= 5+3\) which makes \(x=4\). And for \(2 x - 3 = -5\), \(2x = -5 + 3\) which makes \(x=-1\). Thus we get two possible values for x.
03

Conclusion: Original Statement is False

Looking at the solutions, it's clear that the original statement is false. The square root of 25 can be -5 or 5. Thus, the equation \((2 x - 3)^2 = 25\) is not equivalent to \(2 x - 3 = 5\) but to \(2 x - 3 = 卤 5\).

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