Question

what are the solutions of $x^2 - 7x + 13 = 0$?

a. $x = \frac{7 + 3i}{2}$ or $x = \frac{7 - 3i}{2}$

b. $x = \frac{7 + i\sqrt{2}}{3}$ or $x = \frac{7 - i\sqrt{2}}{3}$

c. $x = 7 + i\sqrt{3}$ or $x = 7 - i\sqrt{3}$

d. $x = \frac{7 + i\sqrt{3}}{2}$ or $x = \frac{7 - i\sqrt{3}}{2}$

Explanation:

Step1: Recall Quadratic Formula

For a quadratic equation \(ax^2 + bx + c = 0\), the solutions are given by \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 1\), \(b=-7\), \(c = 13\).

Step2: Calculate Discriminant (\(D\))

Discriminant \(D=b^2-4ac=(-7)^2-4(1)(13)=49 - 52=-3\).

Step3: Substitute into Quadratic Formula

Since \(D=-3=\sqrt{-1}\times\sqrt{3}=i\sqrt{3}\) (where \(i=\sqrt{-1}\)), we have:
\(x=\frac{-(-7)\pm\sqrt{-3}}{2(1)}=\frac{7\pm i\sqrt{3}}{2}\)

Answer:

D. \(x = \frac{7 + i\sqrt{3}}{2}\) or \(x = \frac{7 - i\sqrt{3}}{2}\)