Question

1. what are the complex solutions to the following equation?

$x^2 - 2x + 3 = 0$
$\bigcirc\\ 1 \pm i\sqrt{2}$
$\bigcirc\\ -1 \pm \sqrt{2}$
$\bigcirc\\ 1 \pm i\sqrt{3}$
$\bigcirc\\ -1 \pm 2\sqrt{i}$

Explanation:

Step1: Identify the quadratic equation form

The given equation is \(x^2 - 2x + 3 = 0\), which is in the standard quadratic form \(ax^2 + bx + c = 0\) where \(a = 1\), \(b = -2\), and \(c = 3\).

Step2: Apply the quadratic formula

The quadratic formula for solving \(ax^2 + bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 1\), \(b = -2\), and \(c = 3\) into the formula:

$$ x=\frac{-(-2)\pm\sqrt{(-2)^2 - 4(1)(3)}}{2(1)}=\frac{2\pm\sqrt{4 - 12}}{2}=\frac{2\pm\sqrt{-8}}{2} $$

Step3: Simplify the square root of a negative number

We know that \(\sqrt{-8}=\sqrt{8}\times\sqrt{-1}=2\sqrt{2}i\) (since \(i = \sqrt{-1}\)). Substitute this back into the expression for \(x\):

$$ x=\frac{2\pm2\sqrt{2}i}{2}=1\pm i\sqrt{2} $$

Answer:

A. \(1 \pm i\sqrt{2}\)