Why do my two methods give different imaginary roots?

[hackedagainanda]

Homework Statement

Solve for x, 7x^2 + 5 = 0

Relevant Equations

$x = \frac {-b \pm \sqrt{b^2 -4ac}} {2a},$

I subtract 5 from both sides to get 7x^2 = -5 Then I divide both sides by 7 to get -5/7. I then take the square root to get x = sqrt of the imaginary unit i 5/7 then $\pm { i \sqrt \frac 5 7}$

The quadratic formula on the other hand gets me a different answer, the discriminant = -140 which can be simplified to 2 sqrt 35 i over 14 and then you factor out the 2 and get $\pm i \sqrt \frac {35} 7$I see there is a Latex error but the root is only in the numerator
Both answers seem correct to me I don't see my error.

 

[berkeman]

b=0, so:
$$x = \pm \frac{\sqrt{-4*35}}{2*7}$$
$$x = \pm \frac{2\sqrt{-35}}{2*7}$$
$$x = \pm \sqrt{\frac{-35}{49}} = \pm \sqrt{\frac{-5}{7}}$$

 

[hackedagainanda]

That's the answer I got, I take it the book rationalized the denominator. I see my error now, I didn't follow the steps all the way through.

 

[hackedagainanda]

Thanks for the help! You are very appreciated