Question

Hello, would it be possible to work through this question so that I could understand it better?

Answer 1

`1\pm\sqrt[]{9}i`

Step-by-step explanation:

The equation we have is:

`(1)/(2)x^2-x+5=0`

And we need to find the solutions to the equation.

Step 1. The first step to solve by completing the square is to modify the given equation to look as follows:

`x^2+bx+c=0`

This means that the x^2 has a coefficient of 1. To do this with our expression, we need to multiply the whole expression by 2:

`2((1)/(2)x^2-x+5=0)`

The result is:

`x^2-2x+10=0`

Step 2. The second step is to write the equation in the form:

`x^2+bx=c`

For this, in our equation, we move the +10 that is on the left side, as a -10 on the right side:

`\begin{gathered} x^2-2x+10=0 \\ \downarrow\downarrow \\ x^2-2x=-10 \end{gathered}`

Comparing this equation with

`x^2+bx=c`

We can find the value of b:

`b=-2`

which we will use to complete the square in step 3.

Step 3. Add (b/2)^2 to both sides of the equation:

`((b)/(2))^2=((-2)/(2))^2=(-1)^2`

we will add this to both sides of our equation:

`\begin{gathered} x^2-2x=-10 \\ \downarrow\downarrow \\ x^2-2x+(-1)^2=-10+(-1)^2 \end{gathered}`

Step 4. The left-hand side can be factored into a perfect square:

`(x-1)^2=-10+(-1)^2`

Step 5. Solving the operations on the right side:

`\begin{gathered} (x-1)^2=-10+1 \\ \downarrow \\ (x-1)^2=-9 \end{gathered}`

And then solving for x:

`\begin{gathered} x-1=\pm√(-9) \\ x=1\pm\sqrt[]{-9} \end{gathered}`

Step 6. Simplify the square root using the definition of ''i'' for the imaginary numbers:

Answer:

`1\pm\sqrt[]{9}i`