Question

find x: (x+3/x-2) - (1-x/x)=17/4.

Answer 1

Hey there!

(x+3/x-2) - (1-x/x)=17/4.

(x+3)x-(1-x)(x-2)/ (x-2)x = 17/4

x²+3x-(x-2-x²+3x)/x²-2x=17/4

x²+3x-x+2+x²-3x/x²-2x=17/4

2x²-x+2/x²-2x=17/4

4(2x²-x+2)=17(x²-2x)

8x²-4x+8=17x²-34x

34x-4x+8=17x²-8x²

30x+8=9x²

9x²-30x-8=0

x= -(-30)±√30²-4(9)(8)/2(9)

=+30±√612/18

=30±6√17/18

=6(5±√17)/18

=5±√17/3


Therefore, x=5+√17/3 or 5-√17/3

Answer 2

$\Large{\underline{\underline{\bf{Solution:-}}}}$

$⟹ \frac{x + 3}{x - 2} - \frac{1 - x}{x} = \frac{17}{4} \\ \\ ⟹\frac{(x)(x + 3) - (x - 2)(1 - x)}{(x - 2)(x)} = \frac{17}{4} \\ \\⟹ \frac{( {x}^{2} + 3x) - (x - {x}^{2} - 2 + 2x) }{ {(x}^{2} - 2x)} = \frac{17}{4} \\ \\⟹ \frac{ {x}^{2} + 3x - x + {x}^{2} + 2 - 2x}{ {x}^{2} - 2x} = \frac{17}{4} \\ \\ ⟹ \frac{2 {x}^{2} + 2 + x }{ {x}^{2} - 2x } = \frac{17}{4}$

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${\bold{\underline{\underline{By\: cross \: multiplication:-}}}}$

$⟹4(2 {x}^{2} + x + 2) = 17( {x}^{2} - 2x) \\ \\⟹ 8 {x}^{2} + 4x + 8 = 17 {x}^{2} - 34x \\ \\⟹ 8 {x}^{2} - 17 {x}^{2} - 4x + 34x + 8 = 0 \\ \\ ⟹ - 9 {x}^{2} + 38x + 8 = 0$

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${\bold{\underline{\underline{By\: middle \: term \: splitting:-}}}}$

$⟹ - 9 {x}^{2} + 38x + 8 = 0 \\ \\ ⟹ - 9 {x}^{2} + 36x + 2x + 8 = 0 \\ \\⟹ - 9x(x + 4) + 2( x + 4) = 0 \\ \\⟹ ( - 9x + 2)(x + 4) = 0$

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${\bold{\underline{\underline{Case-2:-}}}}$

$⟹- 9x + 2 = 0 \\ \\ ⟹ - 9x = - 2 \\ \\⟹ x = \frac{ -2 }{ - 9}$

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${\bold{\underline{\underline{Case-1:-}}}}$

$⟹x + 4 = 0 \\ \\⟹ x = - 4$

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$\Large{\underline{\underline{\bf{Thanks:-}}}}$