Below is one of the gre practice question which i was able to solve but could not understand why my first approach dint worked.
In order to find the value of x, I used the below quadratic equation but it dint worked out well.
$x^2+2x+1=24$
Since $x^2,(x+1),x$ accounts to 24% ($100-76$) of the pie chart.
$x^2+2x-23=0$
Even though the above expression is correct (in context of the question) it does not work well to find the value of x, since above does not have a proper whole-number solutions of x.
On 2nd try i tried below equation and it worked well.
$x^2+2x-76=100$
$x^2+2x-24=0$
$x=-6$ or $x=4$
Now my intent of asking this question on forum is to get a better insight on why my 1st approach dint worked, even though both the equations mentioned above were correct in context of the question.
Actually, your first equation is correct, not your second. You forgot to add the $+1$.
$$(x^2+2x+1)+76 = 100 \implies x^2+2x+77 = 100 \implies x^2+2x = 23$$
This is the first equation again, and since $23$ is prime, this can’t be solved with integers.
$$x^2+2x-23 = 0 \implies x = \frac{-2\pm\sqrt{2^2-4(1)(-23)}}{2(1)} \implies -1\pm 2\sqrt 6$$
Since $-1-2\sqrt 6$ is negative, the answer becomes $-1+2\sqrt 6$%. Slightly awkward, but correct nonetheless...