Math Olympiad Problem | 95% Failed to solve | You should know this Trick !

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[jpharnad]

There is a very simple way to solve this, as follows. Notice that
(10^5)^2 + 10^5 -2, which can be factorized as (10^5 +2)*(10^5 -1). Therefore x^2 +3x can be factorized as (x + 10^5 +2)*(x 10^5 +1), giving the two roots -10^5 -2 and 10^5 -1.

[WhiteGandalfs]

If just inserted into the quadratic solution formular, it simplifies much easier to sqrt( ). I assume that was the original intention of the creator of this problem.
But as always: "All roads lead to Rome"

[NadiehFan]

Your solution is unnecessarily complicated. I solved the problem, of course without first watching your video, in a very simple way.

The equation to solve is

x² + 3x =

Notice that the number at the right hand side is 2 less then 10¹⁰ + 10⁵. At the left hand side we can complete the square by adding (³⁄₂)² = ⁹⁄₄ = 2 + ¹⁄₄ to both sides. This gives us

x² + 3x + (³⁄₂)² = 10¹⁰ + 10⁵ + ¹⁄₄

Now the left hand side x² + 3x + (³⁄₂)² = (x + ³⁄₂)² is a perfect square, but the right hand side is _also_ a perfect square because here we have 10¹⁰ + 10⁵ + ¹⁄₄ = (10⁵)² + 2·(10⁵)·(¹⁄₂) + (¹⁄₂)² = (10⁵ + ¹⁄₂)², so we have

(x + ³⁄₂)² = (10⁵ + ¹⁄₂)²

which gives

x + ³⁄₂ = 10⁵ + ¹⁄₂ ⋁ x + ³⁄₂ = −10⁵ − ¹⁄₂

and so

x = 10⁵ − 1 ⋁ x = −10⁵ − 2

and the equation is solved.