To solve the quadratic equation [tex]\(x^2 + 5x = -2\)[/tex], we need to first rewrite it in the standard form:
[tex]\[ x^2 + 5x + 2 = 0 \][/tex]
The standard form of a quadratic equation is given by:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 5\)[/tex], and [tex]\(c = 2\)[/tex].
To find the solutions, we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, we calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = 5^2 - 4 \cdot 1 \cdot 2 \][/tex]
[tex]\[ \Delta = 25 - 8 \][/tex]
[tex]\[ \Delta = 17 \][/tex]
With the discriminant calculated, we can now find the roots of the equation:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x = \frac{-5 \pm \sqrt{17}}{2} \][/tex]
Thus, the two solutions for the quadratic equation [tex]\(x^2 + 5x + 2 = 0\)[/tex] are:
[tex]\[ x_1 = \frac{-5 + \sqrt{17}}{2} \][/tex]
[tex]\[ x_2 = \frac{-5 - \sqrt{17}}{2} \][/tex]
Among the given options, the solutions [tex]\( \frac{-5 \pm \sqrt{17}}{2} \)[/tex] match with our calculated results. Therefore, the solutions to the equation [tex]\(x^2 + 5x = -2\)[/tex] are:
[tex]\[ \frac{-5 + \sqrt{17}}{2} \text{ and } \frac{-5 - \sqrt{17}}{2} \][/tex]
