To find the solutions to the quadratic equation [tex]\( x^2 - 10x = -34 \)[/tex], we'll follow these steps:
1. Rewrite the equation in standard quadratic form.
First, bring all terms to one side of the equation to set it to zero:
[tex]\[
x^2 - 10x + 34 = 0
\][/tex]
2. Identify coefficients for the quadratic formula.
The standard quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. In this equation:
[tex]\[
a = 1, \quad b = -10, \quad c = 34
\][/tex]
3. Use the quadratic formula.
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substitute in the coefficients [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]
4. Simplify the equation.
Calculate inside the square root:
[tex]\[
x = \frac{10 \pm \sqrt{100 - 136}}{2}
\][/tex]
[tex]\[
x = \frac{10 \pm \sqrt{-36}}{2}
\][/tex]
Simplify the expression under the square root:
[tex]\[
x = \frac{10 \pm \sqrt{36} \cdot i}{2}
\][/tex]
[tex]\[
x = \frac{10 \pm 6i}{2}
\][/tex]
5. Split the equation into two solutions.
[tex]\[
x = \frac{10 + 6i}{2} \quad \text{and} \quad x = \frac{10 - 6i}{2}
\][/tex]
Simplify each fraction:
[tex]\[
x = 5 + 3i \quad \text{and} \quad x = 5 - 3i
\][/tex]
So, the solutions to the equation [tex]\( x^2 - 10x = -34 \)[/tex] are:
[tex]\[
x = 5 \pm 3i
\][/tex]
Therefore, the correct answer is B. [tex]\( x = 5 \pm 3i \)[/tex].
