Question 23064
To solve this problem, which seems to have no rational roots at first, we would have to complete the square. First of all, move 1 to the right side of the equation, then factor out -1 to put it into a more simplified form.
-x^2 - 5x = -1
-(x^2 + 5x) = -1
Then, we need to add a constant to both sides of the equation to obtain a 'perfect square' for the left side of the equation. The constant we need to add is (5/2)^2, since that term would allow 'complete' the square on the left side.
-(x^2 + 5x + 25/4) = -1 - 25/4
Keep in mind that with the sign changed on the left side, 25/4 is subtracted, not added. This expression simplifies to
-(x + 5/2)^2 = -29/4
(x + 5/2)^2 = 29/4
Then take the square root of both sides, and isolate the variable.
x + 5/2 = +- sqrt(29)/2
x = -5/2 +- sqrt(29)/2