Answer:
[tex]x = \frac{ -3 - \sqrt{5} }{2} \: or \: x = \frac{ -3 + \sqrt{5} }{2}[/tex]
Step-by-step explanation:
The given function is
[tex]f(x) = - {x}^{2} - 3x - 1[/tex]
To find the roots, we solve:
[tex] - {x}^{2} - 3x - 1 = 0[/tex]
Multiply through by -1 to get:
[tex] {x}^{2} + 3x + 1 = 0[/tex]
We compare to the general quadratic equation
[tex]a {x}^{2} + bx + c = 0[/tex]
Then we have a=1,b=3, c=1.
The solution is given by:
[tex]x = \frac{ - b\pm \sqrt{ {b}^{2} - 4ac } }{2a} [/tex]
[tex]x = \frac{ - 3\pm \sqrt{ { (- 3)}^{2} - 4 \times1 \times 1 } }{2 \times 1} [/tex]
[tex]x = \frac{ - 3\pm \sqrt{ 9 - 4 } }{2} [/tex]
[tex]x = \frac{ -3\pm \sqrt{5} }{2} [/tex]
[tex]x = \frac{ -3 - \sqrt{5} }{2} \: or \: x = \frac{ -3 + \sqrt{5} }{2} [/tex]
