For this case we must find the solution of the following quadratic equation:
[tex]x ^ 2 + 5x + 7 = 0[/tex]
Where:
[tex]a = 1\\b = 5\\c = 7[/tex]
Then, the solution is given by:
[tex]x = \frac {-b \pm \sqrt {b ^ 2-4 (a) (c)}} {2a}[/tex]
Substituting the values:
[tex]x = \frac {-5 \pm \sqrt {5 ^ 2-4 (1) (7)}} {2 (1)}\\x = \frac {-5 \pm \sqrt {25-28}} {2}\\x = \frac {-5 \pm \sqrt {-3}} {2}[/tex]
By definition we have:[tex]i ^ 2 = -1[/tex]
[tex]x = \frac {-5 \pm \sqrt {3i ^ 2}} {2}\\x = \frac {-5 \pm i \sqrt {3}} {2}[/tex]
Thus, we have two complex roots.
Answer:
The equation has no real roots.
