Question

find all solutions of the equation ( x^2 - 2x + 9 = 0 ) by using the quadratic formula.
( x = )
hint: reduce radicals, no decimal approximations.
question help: video message instructor

Explanation:

Step1: Recall Quadratic Formula

For a quadratic equation \(ax^{2}+bx + c = 0\), the solutions are given by \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). In the equation \(x^{2}-2x + 9 = 0\), we have \(a = 1\), \(b=- 2\), and \(c = 9\).

Step2: Calculate the Discriminant

The discriminant \(D=b^{2}-4ac\). Substituting the values, we get \(D=(-2)^{2}-4\times1\times9=4 - 36=- 32\).

Step3: Substitute into Quadratic Formula

Now, substitute \(a = 1\), \(b=-2\), and \(D=-32\) into the quadratic formula. We know that \(\sqrt{-32}=\sqrt{32}\times\sqrt{-1}=4\sqrt{2}i\) (where \(i=\sqrt{-1}\) is the imaginary unit). So, \(x=\frac{-(-2)\pm\sqrt{-32}}{2\times1}=\frac{2\pm4\sqrt{2}i}{2}\).

Step4: Simplify the Expression

Simplify \(\frac{2\pm4\sqrt{2}i}{2}\) by dividing each term in the numerator by 2. We get \(x = 1\pm2\sqrt{2}i\).

Answer:

\(1 + 2\sqrt{2}i\) and \(1 - 2\sqrt{2}i\) (or written as \(x = 1\pm2\sqrt{2}i\))