(x-4)² + (x+9)² = 2x²

To solve the equation \((x - 4)^2 + (x + 9)^2 = 2x^2\), we will start by expanding the left-hand side:

1. Expand \((x - 4)^2\): \[ (x - 4)^2 = x^2 - 8x + 16 \]

2. Expand \((x + 9)^2\): \[ (x + 9)^2 = x^2 + 18x + 81 \]

3. Now combine the expansions: \[ (x - 4)^2 + (x + 9)^2 = (x^2 - 8x + 16) + (x^2 + 18x + 81) \] Simplifying this gives: \[ = x^2 - 8x + 16 + x^2 + 18x + 81 = 2x^2 + 10x + 97 \]

4. Now set the left-hand side equal to the right-hand side: \[ 2x^2 + 10x + 97 = 2x^2 \]

5. Subtract \(2x^2\) from both sides: \[ 10x + 97 = 0 \]

6. Solve for \(x\): \[ 10x = -97 \] \[ x = -\frac{97}{10} = -9.7 \]

Thus, the solution to the equation \((x - 4)^2 + (x + 9)^2 = 2x^2\) is: \[ \boxed{-9.7} \]