Question

question 3
which two values of x are roots of the polynomial below?
$x^2 + 5x + 9$
a. $x = \frac{-5 + \sqrt{61}}{2}$
b. $x = \frac{5 - \sqrt{17}}{2}$
c. $x = \frac{5 + \sqrt{17}}{2}$
d. $x = \frac{-5 - \sqrt{61}}{2}$
e. $x = \frac{-5 - \sqrt{-11}}{2}$
f. $x = \frac{-5 + \sqrt{-11}}{2}$

Explanation:

Step1: Recall Quadratic Formula

For a quadratic equation \(ax^2 + bx + c = 0\), the roots are given by \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\).
Here, \(a = 1\), \(b = 5\), \(c = 9\).

Step2: Calculate Discriminant (\(D\))

Discriminant \(D=b^2 - 4ac\).
Substitute \(a = 1\), \(b = 5\), \(c = 9\):
\(D=(5)^2-4\times1\times9=25 - 36=- 11\)

Step3: Find Roots Using Formula

Substitute \(a = 1\), \(b = 5\), \(D=-11\) into quadratic formula:
\(x=\frac{-5\pm\sqrt{- 11}}{2}\)
Which gives two roots: \(x=\frac{-5+\sqrt{-11}}{2}\) (Option F) and \(x=\frac{-5-\sqrt{-11}}{2}\) (Option E)

Answer:

E. \(x = \frac{-5 - \sqrt{-11}}{2}\), F. \(x = \frac{-5 + \sqrt{-11}}{2}\)