To solve the expression [tex]\((\frac{x}{2} + \frac{3}{4})^2\)[/tex], we need to expand it step by step.
First, we can recognize that [tex]\((a + b)^2\)[/tex] follows the formula:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
In our case, [tex]\(a = \frac{x}{2}\)[/tex] and [tex]\(b = \frac{3}{4}\)[/tex]. Plugging these into the formula, we get:
[tex]\[
\left(\frac{x}{2} + \frac{3}{4}\right)^2 = \left(\frac{x}{2}\right)^2 + 2 \left(\frac{x}{2}\right) \left(\frac{3}{4}\right) + \left(\frac{3}{4}\right)^2
\][/tex]
Now, let's compute each term separately:
1. [tex]\(\left(\frac{x}{2}\right)^2\)[/tex]:
[tex]\[
\left(\frac{x}{2}\right)^2 = \frac{x^2}{4} = 0.25x^2
\][/tex]
2. [tex]\(2 \left(\frac{x}{2}\right) \left(\frac{3}{4}\right)\)[/tex]:
[tex]\[
2 \left(\frac{x}{2}\right) \left(\frac{3}{4}\right) = 2 \cdot \frac{x}{2} \cdot \frac{3}{4} = \frac{3x}{4} = 0.75x
\][/tex]
3. [tex]\(\left(\frac{3}{4}\right)^2\)[/tex]:
[tex]\[
\left(\frac{3}{4}\right)^2 = \frac{9}{16} = 0.5625
\][/tex]
Now, we can combine all the computed terms:
[tex]\[
\left(\frac{x}{2} + \frac{3}{4}\right)^2 = 0.25x^2 + 0.75x + 0.5625
\][/tex]
Therefore, the expanded form of [tex]\((\frac{x}{2} + \frac{3}{4})^2\)[/tex] is:
[tex]\[
0.25x^2 + 0.75x + 0.5625
\][/tex]
