Question

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Answer 1

1)

Given that , $\sf \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{x} = \dfrac{1}{p + q + x}$.

Need To Find : Value of x ?

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⠀⠀⠀Given that ,

$\dashrightarrow \sf \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{x} = \dfrac{1}{p + q + x} \:$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Solving \: the \: Given \::}}$

$\dashrightarrow \sf \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{x} = \dfrac{1}{p + q + x} \:\\\\ \dashrightarrow \sf \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{x} - \dfrac{1}{p + q + x} = 0 \:\\\\ \dashrightarrow \sf \bigg[ \dfrac{1}{p} + \dfrac{1}{q}\bigg] +\bigg[ \dfrac{1}{x} - \dfrac{1}{p + q + x} \bigg] = 0 \:\\\\ \dashrightarrow \sf \bigg[ \dfrac{p + q }{pq}\bigg] +\bigg[ \dfrac{(x + p + q )- x }{x ( p + q + x ) } \bigg] = 0 \:\\\\\dashrightarrow \sf \bigg[ \dfrac{p + q }{pq}\bigg] +\bigg[ \dfrac{x + p + q - x }{x ( p + q + x ) } \bigg] = 0 \:\\\\\dashrightarrow \sf \bigg[ \dfrac{p + q }{pq}\bigg] +\bigg[ \dfrac{x + p + q - x }{x ( p + q + x ) } \bigg] = 0 \:\\\\ \dashrightarrow \sf \bigg[ \dfrac{p + q }{pq}\bigg] +\bigg[ \dfrac{ p + q }{x ( p + q + x ) } \bigg] = 0 \:\\\\ \dashrightarrow \sf ( p + q ) \bigg[ \dfrac{1}{pq} + \dfrac{ 1 }{x ( p + q + x ) } \bigg] = 0 \:\\\\ \dashrightarrow \sf ( p + q ) \bigg[ \dfrac{ x^2 + xp + xq + pq }{ (pq) (x) ( p + q + x ) } \bigg] = 0 \:\\\\\dashrightarrow \sf ( p + q ) \bigg[ \dfrac{ x^2 + xq + xp + pq }{ (pq) (x) ( p + q + x ) } \bigg] = 0 \:\\\\ \dashrightarrow \sf ( p + q ) \bigg[ \dfrac{ x( x + q ) + p ( x + q ) }{ (pq) (x) ( p + q + x ) } \bigg] = 0 \:\\\\\dashrightarrow \sf ( p + q ) \bigg[ \dfrac{ ( x + p ) ( x + q ) }{ (pq) (x) ( p + q + x ) } \bigg] = 0 \:\\\\ \dashrightarrow \sf \bigg[ \dfrac{ ( x + p ) ( x + q ) }{ (pq) (x) ( p + q + x ) } \bigg] = \dfrac{0}{( p + q )} \:\\\\ \dashrightarrow \sf \bigg[ \dfrac{ ( x + p ) ( x + q ) }{ (pq) (x) ( p + q + x ) } \bigg] = 0 \:\\\\ \dashrightarrow \sf ( x + p ) ( x + q ) = 0 \times (pq) (x) ( p + q + x ) \:\\\\ \dashrightarrow \sf ( x + p ) ( x + q ) = 0 \:\\\\ \dashrightarrow \sf \:x\:=\:-p \:\:or\:\: x \:=\:-q \:$

$\dashrightarrow \:\underline {\boxed{\purple {\pmb{\frak{\:\:x\:=\:-p \:\:or\:\: \:-q \:}}}}}\:\:\bigstar \:\:$

$\qquad \therefore \underline {\sf Hence, The \:value \:of \: x \:can \: be \:\pmb{\bf{ - p \: or \:-q \:}}.}$

2)

Question:-

If ( 2 - √5 )/( 2 + √5 ) =a√5 + b , then find the value of a and b.

To Find:-

Find the value of a and b.

Solution:-

$\dashrightarrow\sf \: \dfrac { 2 - \sqrt { 5 } } { 2 + \sqrt { 5 } } = a \sqrt { 5 } + b$

Rationalise the denominator

$\dashrightarrow\sf \: \dfrac { 2 - \sqrt { 5 } } { 2 + \sqrt { 5 } } \times \dfrac { 2 - \sqrt { 5 } } { 2 - \sqrt { 5 } } = a \sqrt { 5 } + b$

$\dashrightarrow\sf \: \dfrac { { ( 2 - \sqrt { 5 } ) }^{ 2 } } { ( 2 + \sqrt { 5 } )( 2 - \sqrt { 5 } ) } = a \sqrt { 5 } + b$

$\dashrightarrow\sf \: \dfrac { 4 + 5 - 4 \sqrt { 5 } } { 4 - 5 } = a \sqrt { 5 } + b$

$\dashrightarrow\sf \: \dfrac { - ( 9 + 4 \sqrt { 5 } ) } { - 1 } = a \sqrt { 5 } + b$

$\dashrightarrow\sf \: 9 + 4 \sqrt { 5 } = a \sqrt { 5 } + b$

Hence ,

a = 4 , b = 9