Solve 4(a-7)² = (2a-3)²: Finding the Value of a

To solve this problem, we will follow these steps:

• Step 1: Expand both sides of the equation $4(a-7)^2 = (2a-3)^2$.
• Step 2: Simplify the resulting expressions.
• Step 3: Solve for $a$ by equating the simplified expressions.

Now, let's work through each step:

Step 1: Expand both sides.
- Left-side expansion: $4(a-7)^2 = 4(a^2 - 14a + 49) = 4a^2 - 56a + 196$.
- Right-side expansion: $(2a-3)^2 = (2a)^2 - 2 \times 2a \times 3 + 3^2 = 4a^2 - 12a + 9$.

Step 2: Set the expanded expressions equal to each other:
$4a^2 - 56a + 196 = 4a^2 - 12a + 9$.

Now, subtract $4a^2$ from both sides to simplify:
$-56a + 196 = -12a + 9$.

Simplify the equation by bringing all terms involving $a$ to one side and constant terms to the other side:
$-56a + 12a = 9 - 196$.
This simplifies to $-44a = -187$.

Step 3: Solve for $a$ by dividing both sides by $-44$:
$a = \frac{187}{44} = 4\frac{1}{4}$.

Therefore, the solution to the problem is $a = 4\frac{1}{4}$.