Question

Solve the following quadratic equations by factorization: $\frac{x+1}{x-1}-\frac{x-1}{x+1} =\frac{5}{6}$, x≠1, -1

Answer 1

SOLUTION :  

Given : (x + 1)/(x −1) - (x −1)/(x + 1) = 5/6

[ (x + 1) (x + 1) -  x - 1) (x - 1)] /  [(x + 1) (x - 1)] = ⅚  

[ By taking LCM]

[(x + 1)² −(x −1)²] /[x² −1] = ⅚

[(x² + 1² + 2x ) - (x² + 1² - 2x )] / [x² −1] = ⅚

[(a + b)² = a² + b² + 2ab & (a - b)² = a² + b² - 2ab]

[x² + 1² + 2x  - x² - 1² + 2x )] / [x² −1] = ⅚

4x / [x² −1] = ⅚

6(4x) = 5(x² -1)

24x = 5x² - 5

5x² - 24x -5 = 0

5x² - 25x + x- 5 = 0

5x(x - 5) + 1(x - 5) =0

(5x +1)(x - 5) = 0

x - 5 = 0  or  5x + 1 = 0

x = 5  or  5x = -1

x = 5  or x = - 1/5

Hence, the roots of the quadratic equation (x + 1)/(x −1) - (x −1)/(x + 1) = 5/6  are 5 & - ⅕ .

★★ METHOD TO FIND SOLUTION OF a quadratic equation by FACTORIZATION METHOD :  

We first write the given quadratic polynomial as product of two linear factors by splitting the middle term and then equate each factor to zero to get desired roots of given quadratic equation.

HOPE THIS ANSWER WILL HELP YOU….

Answer 2

$\large\boxed{\texttt{\fcolorbox{aqua}{pink}{HØLÁ !!}}}$


$\sf{{\dfrac{x + 1}{x - 1}} - {\dfrac{x - 1}{x + 1}} = {\dfrac{5}{6}}}$


LCM of ( x - 1 ) & ( x + 1 ) = $<b>( x - 1 ) ( x + 1 )</b>$


$\sf{{\dfrac{( x + 1 )^2 - ( x - 1 )^2}{( x - 1 ) ( x + 1 )}} = {\dfrac{5}{6}}}$


$<b>( a + b )² + ( a - b )² = 4ab</b>$

$<b>( a - b ) ( a + b ) = a² - b²</b>$


$\sf{{\dfrac{4 ( x ) ( 1 )}{x^2 - 1}} = {\dfrac{5}{6}}}$


$\sf{{\dfrac{4x}{x^2 - 1}} = {\dfrac{5}{6}}}$


Cross Multiplication

6 ( 4x ) = 5 ( x² - 1 )

24x = 5x² - 5

5x² - 24x - 5 = 0

By Middle Term Factorisation

5x² - 25x + x - 5 = 0

Taking common

5x ( x - 5 ) + 1 ( x - 5 ) = 0

( 5x + 1 ) ( x - 5 ) = 0

Using Zero Product Rule

5x + 1 = 0 and x - 5 = 0

x = -1 / 5 and x = 5

$<b><u>Zeroes are -1 / 5 and 5.</b></u>$