Answer:
[tex] \displaystyle \large \boxed{x = \frac{ - 3 \pm \sqrt{ 7}i }{2} }[/tex]
Step-by-step explanation:
We are given the equation:
[tex] \displaystyle \large{ {x}^{2} + 3x + 4 = 0}[/tex]
Since the expression is not factorable with real numbers, we use the Quadratic Formula.
Quadratic Formula
[tex] \displaystyle \large{x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} }[/tex]
Compare the expression:
[tex] \displaystyle \large{a {x}^{2} + bx + c = {x}^{2} + 3x + 4 }[/tex]
a = 1
b = 3
c = 4
Substitute a = 1, b = 3 and c = 4 in the formula.
[tex] \displaystyle \large{x = \frac{ - 3 \pm \sqrt{ {3}^{2} - 4(1)(4)} }{2(1)} } \\ \displaystyle \large{x = \frac{ - 3 \pm \sqrt{ 9 - 16} }{2} } \\ \displaystyle \large{x = \frac{ - 3 \pm \sqrt{ - 7} }{2} }[/tex]
Imaginary Unit
[tex] \displaystyle \large{i = \sqrt{ - 1} }[/tex]
Therefore,
[tex]\displaystyle \large{x = \frac{ - 3 \pm \sqrt{ 7} \sqrt{ - 1} }{2} } \\ \displaystyle \large{x = \frac{ - 3 \pm \sqrt{ 7}i }{2} }[/tex]
