Question

question 3 of 10
which two values of x are roots of the polynomial below?
x² + 3x + 5

a. x = (-3 - √29)/2
b. x = (-3 + √11)/2
c. x = (-3 - √11)/2
d. x = (-3 - √(-11))/2
e. x = (-3 + √29)/2
f. x = (-3 + √(-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, \(a = 1\), \(b = 3\), \(c = 5\).

Step2: Calculate Discriminant

Discriminant \(D=b^2 - 4ac=(3)^2-4\times1\times5 = 9 - 20=- 11\).

Step3: Find Roots

Substitute \(a\), \(b\), and \(D\) into quadratic formula:
\(x=\frac{-3\pm\sqrt{-11}}{2}\)
Which gives two roots: \(x=\frac{-3+\sqrt{-11}}{2}\) (Option F) and \(x=\frac{-3 - \sqrt{-11}}{2}\) (Option D).

Answer:

D. \(x=\frac{-3-\sqrt{-11}}{2}\), F. \(x=\frac{-3+\sqrt{-11}}{2}\)