Tricky equation

Algebra Level 4

One of the roots of the equation

2000 x 6 + 100 x 5 + 10 x 3 + x − 2 = 0 2000x^{6} + 100x^{5} + 10x^{3} + x - 2 = 0

is of the form

m + n r \frac{ m + \sqrt{n}}{r}

find m + n +r


The answer is 200.

U Z by U Z , 24, Guyana

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1 solution

U Z
Oct 19, 2014

2000 x 6 + ( 100 x 5 + 10 x 3 + x ) − 2 = 0 2000x^{6} + ( 100x^{5} + 10x^{3} + x) - 2 = 0

= 2000 x 6 + x . ( 10 x 2 ) 3 − 1 10 x 2 − 1 − 2 = 0 = 2000x^{6} + x.\dfrac{ (10x^{2})^{3} - 1}{10x^{2} - 1} - 2 =0

x . 1000 x 6 − 1 10 x 2 − 1 = 2 ( 1000 x 6 − 1 ) x.\dfrac{ 1000x^{6} - 1}{10x^{2} - 1} = 2( 1000x^{6} - 1 )

thus

1000 x 6 − 1 = 0 1000x^{6} -1 = 0 or x 10 x 2 − 1 = − 2 \dfrac{x}{10x^{2} -1} = -2

thus

20 x 2 + x − 2 = 0 20x^{2} + x - 2 =0

x = − 1 ± 161 40 x = \dfrac{ -1 \pm \sqrt{161}}{40}

m + n + r = 200 \boxed{m + n +r = 200}

12 Helpful 0 Interesting 0 Brilliant 0 Confused

very nice problem and solution @megh choski

Mardokay Mosazghi - 6 years, 6 months ago

can u explain a bit more.. i dnt get d simplification

Nithin Nithu - 6 years, 2 months ago

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