Solve the Equation: 2a(a-5) = (a+3)² + (a-3)²

To solve this problem, we'll follow these steps:

• Step 1: Expand both squared terms on the right side of the equation
• Step 2: Simplify the terms and combine like terms
• Step 3: Solve the simplified equation for $a$

Let's now work through each step:

Step 1: Expand $(a+3)^2$ and $(a-3)^2$.
We know:
$(a+3)^2 = a^2 + 6a + 9$
$(a-3)^2 = a^2 - 6a + 9$

Step 2: Combine the expansions:
$(a+3)^2 + (a-3)^2 = (a^2 + 6a + 9) + (a^2 - 6a + 9) = 2a^2 + 18$.

Step 3: Now, equate to the left side and simplify:
The left side of the equation is given as $2a(a-5) = 2a^2 - 10a$.

Equating both sides:
$2a^2 - 10a = 2a^2 + 18$

Subtract $2a^2$ from both sides:
$-10a = 18$

Divide by $-10$ to solve for $a$:
$a = \frac{18}{-10} = -1.8$

Therefore, the solution to the problem is $a = -1.8$.