Question

solve the equation.
$x^2 - x + 4 = 0$
$x = \frac{?}{} \pm \frac{\sqrt{}}{}i$
give your answer as a complex number.

Explanation:

Step1: Identify coefficients

For quadratic equation \(ax^2 + bx + c = 0\), here \(a = 1\), \(b=-1\), \(c = 4\).

Step2: Use quadratic formula

Quadratic formula: \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Substitute values: \(x=\frac{-(-1)\pm\sqrt{(-1)^2 - 4(1)(4)}}{2(1)}\).

Step3: Calculate discriminant

Discriminant \(D = b^2 - 4ac=1 - 16=-15\). So \(\sqrt{D}=\sqrt{-15}=\sqrt{15}i\).

Step4: Simplify formula

\(x=\frac{1\pm\sqrt{15}i}{2}\), which can be written as \(x=\frac{1}{2}\pm\frac{\sqrt{15}}{2}i\).

Answer:

\(x = \frac{1}{2} \pm \frac{\sqrt{15}}{2}i\) (So the blanks are filled as: numerator of first fraction \(1\), denominator of first fraction \(2\), numerator of square root \(15\), denominator of the fraction with \(i\) \(2\))