Question

select the correct answer.
what are the solutions of this quadratic equation?
$x^2 - 8x + 97 = 0$
a. $x = 4 + 9i$ and $x = 4 - 9i$
b. $x = -4 + 9i$ and $x = -4 - 9i$
c. $x = -8 + 18i$ and $x = -8 - 18i$
d. $x = 8 + 18i$ and $x = 8 - 18i$

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, \(a = 1\), \(b=-8\), \(c = 97\).

Step2: Calculate discriminant

Discriminant \(D=b^2 - 4ac=(-8)^2-4\times1\times97=64 - 388=- 324\).

Step3: Simplify square root of discriminant

\(\sqrt{D}=\sqrt{-324}=\sqrt{324}\times\sqrt{-1}=18i\) (since \(\sqrt{-1}=i\)).

Step4: Apply quadratic formula

\(x=\frac{-(-8)\pm18i}{2\times1}=\frac{8\pm18i}{2}=4\pm9i\).

Answer:

A. \(x = 4 + 9i\) and \(x = 4 - 9i\)