Question

Complex solutions of x^2+3x+4=0?

Answer 1

Answer:

$\displaystyle \large \boxed{x = \frac{ - 3 \pm \sqrt{ 7}i }{2} }$

Step-by-step explanation:

We are given the equation:

$\displaystyle \large{ {x}^{2} + 3x + 4 = 0}$

Since the expression is not factorable with real numbers, we use the Quadratic Formula.

Quadratic Formula

$\displaystyle \large{x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} }$

Compare the expression:

$\displaystyle \large{a {x}^{2} + bx + c = {x}^{2} + 3x + 4 }$

a = 1

b = 3

c = 4

Substitute a = 1, b = 3 and c = 4 in the formula.

$\displaystyle \large{x = \frac{ - 3 \pm \sqrt{ {3}^{2} - 4(1)(4)} }{2(1)} } \\ \displaystyle \large{x = \frac{ - 3 \pm \sqrt{ 9 - 16} }{2} } \\ \displaystyle \large{x = \frac{ - 3 \pm \sqrt{ - 7} }{2} }$

Imaginary Unit

$\displaystyle \large{i = \sqrt{ - 1} }$

Therefore,

$\displaystyle \large{x = \frac{ - 3 \pm \sqrt{ 7} \sqrt{ - 1} }{2} } \\ \displaystyle \large{x = \frac{ - 3 \pm \sqrt{ 7}i }{2} }$