Question

solve equation by completing the square. list the solutions, separated by commas.\\(2w^2 - 3w + 8 = 0\\)

Explanation:

Step1: Divide by coefficient of \(w^2\)

Divide the entire equation \(2w^{2}-3w + 8=0\) by \(2\) to get \(w^{2}-\frac{3}{2}w + 4 = 0\). Then, rewrite it as \(w^{2}-\frac{3}{2}w=- 4\).

Step2: Complete the square

Take half of the coefficient of \(w\), which is \(\frac{1}{2}\times(-\frac{3}{2})=-\frac{3}{4}\), square it: \((-\frac{3}{4})^{2}=\frac{9}{16}\). Add this to both sides: \(w^{2}-\frac{3}{2}w+\frac{9}{16}=-4+\frac{9}{16}\).
The left side becomes \((w - \frac{3}{4})^{2}\), and the right side is \(\frac{-64 + 9}{16}=\frac{-55}{16}\).

Step3: Take square roots

Take the square root of both sides: \(w-\frac{3}{4}=\pm\sqrt{\frac{-55}{16}}=\pm\frac{\sqrt{55}i}{4}\) (since \(\sqrt{-1}=i\)).

Step4: Solve for \(w\)

Add \(\frac{3}{4}\) to both sides: \(w=\frac{3}{4}\pm\frac{\sqrt{55}i}{4}=\frac{3\pm\sqrt{55}i}{4}\).

Answer:

\(\frac{3 + \sqrt{55}i}{4},\frac{3 - \sqrt{55}i}{4}\)