Question

question 2 which two values of x are roots of the polynomial below? $x^2 + 5x + 7$ a. $x = \frac{5 - \sqrt{17}}{2}$ b. $x = \frac{5 + \sqrt{17}}{2}$ c. $x = 5$ d. $x = \frac{1}{2}$ e. $x = \frac{-5 - \sqrt{-3}}{2}$ f. $x = \frac{-5 + \sqrt{-3}}{2}$

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, the polynomial is \(x^{2}+5x + 7\), so \(a = 1\), \(b = 5\), \(c = 7\).

Step2: Calculate Discriminant

Discriminant \(D=b^{2}-4ac=(5)^{2}-4\times1\times7=25 - 28=- 3\).

Step3: Find Roots

Substitute \(a\), \(b\), and \(D\) into the quadratic formula: \(x=\frac{-5\pm\sqrt{-3}}{2}\). So the two roots are \(x=\frac{-5-\sqrt{-3}}{2}\) and \(x=\frac{-5 + \sqrt{-3}}{2}\).

Answer:

E. \(x=\frac{-5-\sqrt{-3}}{2}\), F. \(x=\frac{-5+\sqrt{-3}}{2}\)