Question

If using the method of completing the square to solve the quadratic equation x2 + x +9 = 0, which number would have to be added to "complete the square"?

Answer 1

The given equation is

`x^2+x+9=0`

To complete the square, first, we have to move the independent term to the other side.

`x^2+x=-9`

Then, we divide the linear coefficient by 2, then we apply a square power to it.

`((1)/(2))^2=(1)/(4)`

We add this fraction to each side.

`x^2+x+(1)/(4)=-9+(1)/(4)`

Then, we factor the trinomial and sum the numbers on the right side.

`\begin{gathered} (x+(1)/(2))^2=(-36+1)/(4) \\ (x+(1)/(2))^2=(-35)/(4) \end{gathered}`

Then, to solve for x, we use a square root on both sides.

`\begin{gathered} \sqrt[]{(x+(1)/(2))^2}=\pm\sqrt[]{-(35)/(4)} \\ (x+(1)/(2))=\pm\sqrt[]{-(35)/(4)} \end{gathered}`

As you can observe, the equation has no real solutions because the square root of a negative number can be solved in the real numbers.

Hence, the equation has no real solutions, and the number added to complete the square is 1/4.