Certainly! Let's solve the problem step by step.
Given that one of the roots of the equation [tex]\(2x^2 - k(x + 1) + 3 = 0\)[/tex] is [tex]\(4\)[/tex], we can substitute [tex]\(x = 4\)[/tex] into the given equation to find [tex]\(k\)[/tex].
### Step-by-Step Solution:
1. Substitute the root into the equation.
Since [tex]\(4\)[/tex] is a root, it must satisfy the equation:
[tex]\[
2x^2 - k(x + 1) + 3 = 0
\][/tex]
Substitute [tex]\(x = 4\)[/tex]:
[tex]\[
2(4)^2 - k(4 + 1) + 3 = 0
\][/tex]
2. Simplify the equation.
Calculate [tex]\(2(4)^2\)[/tex]:
[tex]\[
2 \cdot 16 = 32
\][/tex]
Calculate [tex]\(k(4 + 1)\)[/tex]:
[tex]\[
k \cdot 5 = 5k
\][/tex]
Therefore, the equation becomes:
[tex]\[
32 - 5k + 3 = 0
\][/tex]
3. Combine the constant terms.
Add [tex]\(32\)[/tex] and [tex]\(3\)[/tex]:
[tex]\[
35 - 5k = 0
\][/tex]
4. Solve for [tex]\(k\)[/tex].
Isolate [tex]\(k\)[/tex] by moving [tex]\(35\)[/tex] to the other side:
[tex]\[
-5k = -35
\][/tex]
Divide both sides by [tex]\(-5\)[/tex]:
[tex]\[
k = 7
\][/tex]
Therefore, the value of [tex]\(k\)[/tex] that satisfies the given equation with a root of [tex]\(4\)[/tex] is:
[tex]\[
k = 7
\][/tex]
That's the complete solution. [tex]\(k = 7\)[/tex].
