1.3. Given \( (x-3)^{\frac{2}{2}}=p^{2}-4 \) 1.3.1. Determine the values of P for whi

To determine the values of \( p \) for which the equation \( (x-3)^{\frac{2}{2}}=p^{2}-4 \) holds true, we need to solve the equation for \( p \). Given equation: \( (x-3)^{\frac{2}{2}}=p^{2}-4 \) Since \( (x-3)^{\frac{2}{2}} \) simplifies to \( x-3 \), the equation becomes: \[ x-3=p^{2}-4 \] Now, we can solve for \( p \) by isolating it on one side of the equation. Solve the equation by following steps: - step0: Solve for \(p\): \(x-3=p^{2}-4\) - step1: Swap the sides: \(p^{2}-4=x-3\) - step2: Move the constant to the right side: \(p^{2}=x-3+4\) - step3: Add the numbers: \(p^{2}=x+1\) - step4: Simplify the expression: \(p=\pm \sqrt{x+1}\) - step5: Separate into possible cases: \(\begin{align}&p=\sqrt{x+1}\\&p=-\sqrt{x+1}\end{align}\) The values of \( p \) for which the equation \( (x-3)^{\frac{2}{2}}=p^{2}-4 \) holds true are \( p=\sqrt{x+1} \) and \( p=-\sqrt{x+1} \).