(x+10)²=(5-x)²
To solve the equation \((x+10)²=(5-x)²\), we'll start by expanding both sides.
Expand the left side: \[ (x+10)² = x² + 20x + 100 \]
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: \[ x² + 20x + 100 = 25 - 10x + x² \]
Cancel \(x²\) from both sides: \[ 20x + 100 = 25 - 10x \]
Now, rearrange the equation to isolate \(x\): \[ 20x + 10x = 25 - 100 \] \[ 30x = -75 \]
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.
