Test of Mathematics Solution Subjective 61 - Symmetric Polynomial

This is a Test of Mathematics Solution Subjective 61 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

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Problem

Solve ${{6x}^{2}}$ - 25x + 12 + ${\frac{25}{x}}$ + ${\frac{6}{x^2}}$ = 0.

Solution

${{6x}^{2}}$ – 25x + 12 + ${\frac{25}{x}}$ + ${\frac{6}{x^2}}$ = 0
= 6( ${x^2}$ + ${\frac{1}{x^2}}$ ) – 25(x - ${\frac{1}{x}}$ ) +12 = 0
= 6 (x – ${\frac{1}{x}})^2$ – 25 (x – ${\frac{1}{x}}$ ) + 24 = 0
= (x – ${\frac{1}{x}}$ ) = ${\frac{25\pm {\sqrt49}}{12}}$ = ${\frac{3}{2}}$ or ${\frac{8}{3}}$
If (x – ${\frac{1}{x}}$ ) = ${\frac{3}{2}}$
Or ${{2x}^2}$ – 3x – 2 = 0
Or x = 2 , - ${\frac{1}{2}}$
If (x – ${\frac{1}{x}}$ ) = ${\frac{8}{3}}$
Or ${x^2}$ – 1 = ${\frac{8}{3x}}$
Or ${3x^2}$ – 8x – 3 = 0
Or x = 3, -1/3
X = 2,- ${\frac{1}{2}}$ ,3,- ${\frac{1}{3}}$

Problem

Solve ${{6x}^{2}}$ - 25x + 12 + ${\frac{25}{x}}$ + ${\frac{6}{x^2}}$ = 0.

Solution

${{6x}^{2}}$ – 25x + 12 + ${\frac{25}{x}}$ + ${\frac{6}{x^2}}$ = 0
= 6( ${x^2}$ + ${\frac{1}{x^2}}$ ) – 25(x - ${\frac{1}{x}}$ ) +12 = 0
= 6 (x – ${\frac{1}{x}})^2$ – 25 (x – ${\frac{1}{x}}$ ) + 24 = 0
= (x – ${\frac{1}{x}}$ ) = ${\frac{25\pm {\sqrt49}}{12}}$ = ${\frac{3}{2}}$ or ${\frac{8}{3}}$
If (x – ${\frac{1}{x}}$ ) = ${\frac{3}{2}}$
Or ${{2x}^2}$ – 3x – 2 = 0
Or x = 2 , - ${\frac{1}{2}}$
If (x – ${\frac{1}{x}}$ ) = ${\frac{8}{3}}$
Or ${x^2}$ – 1 = ${\frac{8}{3x}}$
Or ${3x^2}$ – 8x – 3 = 0
Or x = 3, -1/3
X = 2,- ${\frac{1}{2}}$ ,3,- ${\frac{1}{3}}$