Question

If using the method of completing the square to solve the quadratic equation x^2 + 3x +21 =0?

Answer 1

`x= \frac 12 \left( -3 +i5\sqrt 3\right)\\\\x= \frac 12 \left( -3 -i5\sqrt 3\right)`

Explanation:

`~~~~~~x^2 +3x +21 = 0\\\\\implies x^2 +3x = -21\\\\\implies x^2 + 2\cdot \frac 32 \cdot x + \left( \frac 32 \right)^2 = -21 + \left( \frac 32 \right)^2\\\\\implies \left(x + \frac 32 \right)^2 = -21+\frac 94\\\\\implies \left(x + \frac 32 \right)^2 = -\frac{75}4\\\\\implies x+ \frac 32 = \pm\sqrt{-\frac{75}4 \right)\\\\\implies x + \frac 32 = \pm i (5\sqrt 3)/(2)\\\\\implies x = -\frac 32 \pm i \frac {5\sqrt 3}2\\\\\implies x = \frac 12 \left( -3 \pm i5\sqrt 3\right)`