Problem 170 - College Algebra

Question

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 \(2 x-3=5\)

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: 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 \(2 x - 3 = ± 5\).