Sure! Let's go through the steps to simplify the given expression:
[tex]\[
\frac{1}{2} - 3\left(\frac{1}{2} + 1\right)^2
\][/tex]
### Step 1: Simplify inside the parenthesis
First, simplify the expression inside the parenthesis:
[tex]\[
\frac{1}{2} + 1
\][/tex]
To add these fractions, we need a common denominator. We can rewrite [tex]\(1\)[/tex] as [tex]\(\frac{2}{2}\)[/tex]:
[tex]\[
\frac{1}{2} + \frac{2}{2} = \frac{3}{2}
\][/tex]
### Step 2: Square the simplified inner parenthesis
Next, we square the result from the first step:
[tex]\[
\left(\frac{3}{2}\right)^2
\][/tex]
Squaring a fraction means squaring the numerator and the denominator:
[tex]\[
\left(\frac{3}{2}\right)^2 = \frac{3^2}{2^2} = \frac{9}{4}
\][/tex]
### Step 3: Multiply by -3
Now, multiply the squared term by 3:
[tex]\[
3 \times \frac{9}{4}
\][/tex]
This can be done by multiplying the numerator:
[tex]\[
3 \times \frac{9}{4} = \frac{27}{4}
\][/tex]
### Step 4: Subtract from [tex]\(\frac{1}{2}\)[/tex]
Finally, subtract this result from [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
\frac{1}{2} - \frac{27}{4}
\][/tex]
First, rewrite [tex]\(\frac{1}{2}\)[/tex] with the same denominator as [tex]\(\frac{27}{4}\)[/tex]:
[tex]\[
\frac{1}{2} = \frac{2}{4}
\][/tex]
Now we have:
[tex]\[
\frac{2}{4} - \frac{27}{4} = \frac{2 - 27}{4} = \frac{-25}{4}
\][/tex]
So, the simplified form of the expression is:
[tex]\[
\boxed{\frac{-25}{4}}
\][/tex]
