Steps:
So for this, I will be completing the square. Firstly, subtract 5 on both sides of the equation:
[tex]\frac{1}{2}x^2-x=-5[/tex]
Next, multiply both sides by 2:
[tex]x^2-2x=-10[/tex]
Next, we want to make the left side of the equation a perfect square. To find the constant of the soon-to-be perfect square, divide the x coefficient by 2 and then square the quotient. Add the result to both sides of the equation:
-2 ÷ 2 = -1, (-1)² = 1
[tex]x^2-2x+1=-9[/tex]
Next, factor the left side:
[tex](x-1)^2=-9[/tex]
Next, square root both sides of the equation:
[tex]x-1=\pm \sqrt{-9}[/tex]
Next, remember that i = √-1 so factor that out on the right side of the equation:
[tex]x-1=\pm i\sqrt{9}[/tex]
Lastly, add both sides by 1:
[tex]x=1\pm i\sqrt{9}[/tex]
Answer:
In short, your answer is [tex]x=1\pm i\sqrt{9}[/tex] , or the third option.
