To evaluate the expression [tex]\(-2\left|x^2 - 15\right| - 4\)[/tex] when [tex]\(x = 83\)[/tex], follow these steps:
1. Substitute the value of [tex]\(x\)[/tex] into the expression:
Replace [tex]\(x\)[/tex] with 83 in the expression to get:
[tex]\[
-2\left|83^2 - 15\right| - 4
\][/tex]
2. Calculate [tex]\(83^2\)[/tex]:
[tex]\[
83^2 = 6889
\][/tex]
3. Compute [tex]\(83^2 - 15\)[/tex]:
Subtract 15 from 6889:
[tex]\[
6889 - 15 = 6874
\][/tex]
4. Determine the absolute value [tex]\(\left|6874\right|\)[/tex]:
Since 6874 is positive, its absolute value is:
[tex]\[
\left|6874\right| = 6874
\][/tex]
5. Multiply by -2:
Multiply 6874 by -2:
[tex]\[
-2 \times 6874 = -13748
\][/tex]
6. Subtract 4 from the result:
Finally, subtract 4 from -13748 to get:
[tex]\[
-13748 - 4 = -13752
\][/tex]
Thus, the value of the expression [tex]\(-2\left|x^2-15\right|-4\)[/tex] when [tex]\(x = 83\)[/tex] is [tex]\(-13752\)[/tex].
Since none of the provided options [tex]\(a)\)[/tex] -16, [tex]\(b)\)[/tex] -52, [tex]\(c)\)[/tex] 44, and [tex]\(d)\)[/tex] 8 match this value, the correct answer is not listed among the options. The final evaluated expression is [tex]\( -13752 \)[/tex].
