To rewrite the given equation [tex]\(2x^2 - 8x = 10\)[/tex] in the form [tex]\(2(x - p)^2 + q = 0\)[/tex] and determine the value of [tex]\(q\)[/tex], follow these steps:
1. Move the constant term to the left side of the equation:
[tex]\[
2x^2 - 8x - 10 = 0
\][/tex]
2. Divide the equation by 2 to simplify it:
[tex]\[
x^2 - 4x - 5 = 0
\][/tex]
3. Complete the square for the quadratic expression [tex]\(x^2 - 4x\)[/tex]:
- First, take the coefficient of [tex]\(x\)[/tex], which is [tex]\(-4\)[/tex], and divide it by 2, giving [tex]\(-2\)[/tex].
- Next, square [tex]\(-2\)[/tex] to get [tex]\(4\)[/tex].
So, add and subtract [tex]\(4\)[/tex] inside the equation:
[tex]\[
x^2 - 4x + 4 - 4 - 5 = 0
\][/tex]
This simplifies to:
[tex]\[
(x - 2)^2 - 9 = 0
\][/tex]
4. Return to the original form with the [tex]\(2\)[/tex] factor:
[tex]\[
2(x - 2)^2 - 18 = 10
\][/tex]
5. Move [tex]\(-18\)[/tex] to the right side of the equation to solve for the desired form:
[tex]\[
2(x - 2)^2 - 18 + 18 = 10 + 18
\][/tex]
[tex]\[
2(x - 2)^2 + 28 = 0
\][/tex]
Thus, the value of [tex]\(q\)[/tex] is:
[tex]\[
\boxed{28}
\][/tex]
