To solve for the values of [tex]\( x \)[/tex] such that [tex]\( \|v + w\| = 4 \)[/tex] where [tex]\( v = 2i - 5j \)[/tex] and [tex]\( w = xi + 7j \)[/tex], follow these steps:
1. Express the vectors in component form:
- [tex]\( v = (2, -5) \)[/tex]
- [tex]\( w = (x, 7) \)[/tex]
2. Compute the sum of the vectors [tex]\( v + w \)[/tex]:
[tex]\[
v + w = (2 + x)i + (-5 + 7)j = (2 + x, 2)
\][/tex]
3. Calculate the magnitude of the resulting vector [tex]\( v + w \)[/tex]:
[tex]\[
\|v + w\| = \sqrt{(2 + x)^2 + 2^2}
\][/tex]
4. Set the magnitude equal to 4 and form the equation:
[tex]\[
\sqrt{(2 + x)^2 + 4} = 4
\][/tex]
5. Square both sides to eliminate the square root:
[tex]\[
(2 + x)^2 + 4 = 16
\][/tex]
6. Simplify the equation:
[tex]\[
(2 + x)^2 + 4 = 16 \implies (2 + x)^2 = 12
\][/tex]
7. Solve for [tex]\( 2 + x \)[/tex]:
[tex]\[
2 + x = \pm \sqrt{12} \implies 2 + x = \pm 2\sqrt{3}
\][/tex]
8. Isolate [tex]\( x \)[/tex]:
[tex]\[
x = -2 + 2\sqrt{3} \quad \text{or} \quad x = -2 - 2\sqrt{3}
\][/tex]
Therefore, the solutions for [tex]\( x \)[/tex] are:
[tex]\[
x = -2 + 2\sqrt{3}, \quad x = -2 - 2\sqrt{3}
\][/tex]
Hence, the correct choice is:
A. [tex]\( x = -2 + 2\sqrt{3}, -2 - 2\sqrt{3} \)[/tex]
