To solve the quadratic equation [tex]\(x^2 - 10x = -34\)[/tex], we first need to write it in standard form [tex]\(ax^2 + bx + c = 0\)[/tex].
Step 1: Rearrange the equation to standard form.
[tex]\[ x^2 - 10x + 34 = 0 \][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = 34\)[/tex].
Step 2: Use the quadratic formula to find the roots.
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
Step 3: Plug in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
[tex]\[ x = \frac{{-(-10) \pm \sqrt{{(-10)^2 - 4 \cdot 1 \cdot 34}}}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{{10 \pm \sqrt{{100 - 136}}}}{2} \][/tex]
[tex]\[ x = \frac{{10 \pm \sqrt{{-36}}}}{2} \][/tex]
Step 4: Simplify the expression under the square root.
[tex]\[ x = \frac{{10 \pm \sqrt{{-36}}}}{2} \][/tex]
[tex]\[ \sqrt{{-36}} = 6i \][/tex]
Step 5: Substitute [tex]\(\sqrt{-36}\)[/tex] with [tex]\(6i\)[/tex].
[tex]\[ x = \frac{{10 \pm 6i}}{2} \][/tex]
[tex]\[ x = \frac{10}{2} \pm \frac{6i}{2} \][/tex]
[tex]\[ x = 5 \pm 3i \][/tex]
Step 6: Write down the solutions.
[tex]\[ x = 5 + 3i \][/tex]
[tex]\[ x = 5 - 3i \][/tex]
Therefore, the solutions of the quadratic equation [tex]\(x^2 - 10x + 34 = 0\)[/tex] are
[tex]\[ x = 5 \pm 3i \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{B. \; x = 5 \pm 3i} \][/tex]
