Solve (7+a)² = (½a+8)² + ¾a²: Finding the Value of a

To solve this problem, follow these steps:

• Step 1: Expand both sides of the equation.
• Step 2: Simplify and combine like terms.
• Step 3: Solve the quadratic equation for $a$.

Step 1: Expand both sides:

The left side of the equation is $(7+a)^2$ which expands to:

$(7+a)^2 = 7^2 + 2 \cdot 7 \cdot a + a^2 = 49 + 14a + a^2$.

The right side of the equation is $\left(\frac{1}{2}a + 8\right)^2 + \frac{3}{4}a^2$. First, expand the square:

$\left(\frac{1}{2}a + 8\right)^2 = \left(\frac{1}{2}a\right)^2 + 2 \cdot \frac{1}{2}a \cdot 8 + 8^2$.

$= \frac{1}{4}a^2 + 8a + 64$.

Thus, the right side becomes:

$\frac{1}{4}a^2 + 8a + 64 + \frac{3}{4}a^2$.

$= a^2 + 8a + 64$.

Step 2: Set the expanded equations equal and simplify:

$49 + 14a + a^2 = a^2 + 8a + 64$.

Cancel $a^2$ from both sides:

$49 + 14a = 8a + 64$.

Rearrange terms to isolate $a$:

$14a - 8a = 64 - 49$.

$6a = 15$.

Step 3: Solve for $a$:

$a = \frac{15}{6} = \frac{5}{2}$.

Therefore, the solution to the problem is $a = 2\frac{1}{2}$.