Question

question 10 of 10
which two values of x are roots of the polynomial below?
$5x^2 - 5x + 1$

a. $x = \frac{5 + \sqrt{45}}{20}$
b. $x = \frac{5 + \sqrt{5}}{10}$
c. $x = \frac{-8 + \sqrt{28}}{6}$
d. $x = \frac{-8 - \sqrt{28}}{6}$
e. $x = \frac{5 - \sqrt{45}}{20}$
f. $x = \frac{5 - \sqrt{5}}{10}$

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, \(a = 5\), \(b=-5\), \(c = 1\).

Step2: Calculate Discriminant

Discriminant \(D=b^2-4ac=(-5)^2-4\times5\times1=25 - 20 = 5\).

Step3: Apply Quadratic Formula

Substitute \(a\), \(b\), \(D\) into the formula:
\(x=\frac{-(-5)\pm\sqrt{5}}{2\times5}=\frac{5\pm\sqrt{5}}{10}\).

Answer:

B. \(x = \frac{5+\sqrt{5}}{10}\), F. \(x = \frac{5-\sqrt{5}}{10}\)