Here is a detailed, step-by-step solution for the given expression evaluated at [tex]\( a = 8 \)[/tex]:
### Step 1:
Evaluate the expression with [tex]\( a = 8 \)[/tex]:
[tex]\[
-\frac{1}{2}(-4(8)-6) + 8^2
\][/tex]
### Step 2:
Calculate the value inside the parentheses:
[tex]\[
-4(8) = -32
\][/tex]
Thus, the expression becomes:
[tex]\[
-\frac{1}{2}(-32 - 6) + 8^2
\][/tex]
### Step 3:
Combine the terms inside the parentheses:
[tex]\[
-32 - 6 = -38
\][/tex]
So the expression simplifies to:
[tex]\[
-\frac{1}{2}(-38) + 8^2
\][/tex]
### Step 4:
Evaluate the multiplication by [tex]\(-\frac{1}{2}\)[/tex]:
[tex]\[
-\frac{1}{2}(-38) = 19
\][/tex]
Now, the expression is:
[tex]\[
19 + 8^2
\][/tex]
### Step 5:
Calculate the square of 8:
[tex]\[
8^2 = 64
\][/tex]
So the expression updates to:
[tex]\[
19 + 64
\][/tex]
### Step 6:
Add the results:
[tex]\[
19 + 64 = 83
\][/tex]
Leon claims the following steps, but upon analysis, here is the correct evaluation:
1. [tex]$-\frac{1}{2}(-4(8)-6)+8^2 = 83$[/tex]
2. [tex]$-\frac{1}{2}(-32-6)+8^2 = 83$[/tex]
3. [tex]$-\frac{1}{2}(-38)+8^2 = 83$[/tex]
4. [tex]$-\frac{1}{2}(-38)+16 = 83$[/tex]
5. [tex]$-19+16$[/tex] does not apply since the correct answer up to this point is [tex]\(83\)[/tex].
The correct answer to the expression [tex]\( -\frac{1}{2}(-4a - 6) + a^2 \)[/tex] evaluated at [tex]\( a = 8 \)[/tex] is indeed [tex]\( 83 \)[/tex]. Therefore, step 6 must reflect a valid evaluation, which in our case ends at step 4 successfully.
So, the correct sequence of steps leads to the final value of [tex]\( 83 \)[/tex] directly.
