Question

what are the solutions of $x^2 - 2x + 26 = 0$?

a. $x = 2 + 10i$ or $x = 2 - 10i$

b. $x = 1 + 5i$ or $x = 1 - 5i$

c. $x = 2 + 5i$ or $x = 2 - 5i$

d. $x = 5i$ or $x = -5i$

Explanation:

Step1: Recall quadratic formula

For \(ax^2 + bx + c = 0\), solutions are \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 1\), \(b=-2\), \(c = 26\).

Step2: Calculate discriminant

Discriminant \(D=b^2 - 4ac=(-2)^2-4(1)(26)=4 - 104=-100\).

Step3: Simplify square root of discriminant

\(\sqrt{D}=\sqrt{-100}=10i\) (since \(\sqrt{-1}=i\)).

Step4: Apply quadratic formula

\(x=\frac{-(-2)\pm10i}{2(1)}=\frac{2\pm10i}{2}=1\pm5i\).

Answer:

B. \(x = 1 + 5i\) or \(x = 1 - 5i\)