(x+10)²=(5-x)²

To solve the equation \((x+10)²=(5-x)²\), we'll start by expanding both sides. 1. Expand the left side: \[ (x+10)² = x² + 20x + 100 \] 2. Expand the right side: \[ (5-x)² = (5)² - 2 \cdot 5 \cdot x + x² = 25 - 10x + x² \] Now, set the expanded sides equal to each other: \[ x² + 20x + 100 = 25 - 10x + x² \] 3. Cancel \(x²\) from both sides: \[ 20x + 100 = 25 - 10x \] 4. Now, rearrange the equation to isolate \(x\): \[ 20x + 10x = 25 - 100 \] \[ 30x = -75 \] 5. Divide both sides by 30: \[ x = \frac{-75}{30} = -\frac{5}{2} = -2.5 \] Thus, the solution is: \[ \boxed{-2.5} \] To verify, we can substitute \(x = -2.5\) back into the original equation: Left side: \[ (-2.5 + 10)² = (7.5)² = 56.25 \] Right side: \[ (5 - (-2.5))² = (5 + 2.5)² = (7.5)² = 56.25 \] Both sides equal \(56.25\), confirming that the solution \(x = -2.5\) is correct.