Problem 170 - Algebra and Trigonometry

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

False, the correct equation should be '(2x-3)^{2}=25' is equivalent to '2x-3=5' and '2x-3=-5'.

Step 1: Equation Analysis

The given equation is \((2x-3)^{2}=25\), and it's claimed to be equivalent to \(2x-3=5\).

Step 2: Taking the square root of both sides

Since we have \(a^{2}=b\), this implies that \(a\) can be \(\sqrt{b}\) or \(-\sqrt{b}\). So, taking the square root of both sides of the equation \((2 x-3)^{2}=25\), we get \(2x - 3 = -5\) or \(2x - 3 = 5\).

Step 3: Solving for x

Now, solving the equations \(2x-3=-5\) and \(2x-3=5\) will give us the solution for \(x\). For \(2x-3=-5\), the solution is \(x=-1\), and for \(2x-3=5\), the solution is \(x=4\).

Step 4: Conclusion

The equation \((2x-3)^{2} = 25\) actually gives two solutions for \(x\), namely \(x=-1\) and \(x=4\). This means that the given statement 'The equation \((2x-3)^{2}=25\) is equivalent to \(2 x-3=5\)' is false because it only represents one of the two solutions of the equation. The correct statement should be 'The equation \((2x-3)^{2}=25\) is equivalent to \(2x-3=5\) and \(2x-3=-5\)'.