Question

Solve quadratic equation :

$\frac{x + 3}{x - 2} - \frac{1 - x}{x} = 4 \frac{1}{4}$


, x # 2 , 0

Answer 1

HEY THERE!!!


Question;-

Solve quadratic equation :

$\frac{x + 3}{x - 2} - \frac{1 - x}{x} = 4 \frac{1}{4}$

, x # 2 , 0

Method of Solution;-

$\frac{x + 3}{x - 2} - \frac{1 - x}{x} = 4 \frac{1}{4} \\ \\ \\ \frac{x(x + 3) - (x - 2)(1 - x)}{(x - 2)(x)} = \frac{17}{4} \\ \\ \\ \frac{ {x}^{2} + 3x - \: (x - {x}^{2} - 2 + 2x) }{ {x}^{2} - 2x } = \frac{17}{4} \\ \\ \frac{ {x}^{2} + 3x - \: (3x - {x}^{2} - 2 ) }{ {x}^{2} - 2x } \\ \\ \\ \frac{ {x}^{2} + 3x - 3x + {x}^{2} + 2}{ {x}^{2} - 2x } \\ \\ \\ \\ \frac{ {2x}^{2} + 2 }{ {x}^{2} - 2x} = \frac{17}{4} \\ \\ \\ 4( {2x}^{2} + 2) = 17( {x}^{2} - 2x) \\ \\ {8x}^{2} + 8 = {17x}^{2} - 34x \\ \\ \\ {17x}^{2} - 34x - {8x }^{2} = 8 \\ \\ \\ {9x}^{2} - 34x = 8 \\ \\ \\ {9x}^{2} - 34x - 8 = 0 \\ \\ \\ {9x}^{2} - 36x + 2x - 8 = 0 \\ \\ 9x(x - 4) + 2(x - 4) = 0 \\ \\ (9x + 2)(x - 4) = 0 \\ \\ \\ 9x + 2 = 0 \\ \\ 9x = - 2 \\ \\ x = \frac{ - 2}{9} \\ \\ \mathsf{ other \: roots \: for \: the \: quardratic \: equation} \\ \\ x - 4 = 0 \\ \\ x = 4$


Hence,
$\frac{x + 3}{x - 2} - \frac{1 - x}{x} = 4 \frac{1}{4}$ roots are -2/9 and 4.

Answer 2

Given equation : $\dfrac{x+3}{x-2}+\dfrac{1-x}{x} = 4\dfrac{1}{4}$

$\mathsf{\implies \bigg\{\dfrac{x+3}{x-2} \times \dfrac{x}{x} \bigg\} - \bigg\{ \dfrac{1-x}{x} \times\dfrac{x-2}{x-2} \bigg\} = \dfrac{17}{4}}$

$\mathsf{ \implies \dfrac{x(x + 3 ) }{x( x - 2 ) } - \dfrac{( 1-x)(x-2)}{x(x-2)}= \dfrac{17}{4} }$

$\mathsf{\implies \dfrac{x^2 + 3x - \{1(x-2) - x(x-2)\} }{x^2-2x} =\dfrac{17}{4}} \\ \\ \\ \mathsf{\implies\dfrac{x^2 + 3x - \{x-2-x^2+2x\}}{x^2-2x} =\dfrac{17}{4}} \\\\\\\mathsf{\implies\dfrac{x^2+3x-\{-x^2+3x-2\}}{x^2-2x}=\dfrac{17}{4} }\\\\\\\mathsf{\implies\dfrac{x^2+3x+x^2-3x+2}{x^2-2x}=\dfrac{17}{4}}$

$\mathsf{\implies \dfrac{2x^2+2}{x^2-2x}=\dfrac{17}{4} }\\\\\\\mathsf{\implies8x^2 + 8 = 17x^2-34x}\\\\\\\mathsf{\implies17x^2-8x^2-34x-8 = 0 }\\\\\\\mathsf{\implies 9x^2 - 34x -8=0}\\\\\\\mathsf{\implies 9x^2-(36-2)x- 8 = 0 }$

⇒ 9x^2 - 36x + 2x - 8 = 0

⇒ 9x( x - 4 ) + 2( x - 4 ) = 0

⇒ ( x - 4 )( 9x + 2 ) = 0

⇒ x = 4 or x = - 2 / 9