Question

$Given~that~p+q=x ~ \\\\ where~p= \bigg(x-\frac{1}{x}\bigg)^{\frac{1}{2}}~and~q= \bigg(1-\frac{1}{x}\bigg)^{\frac{1}{2}}\\\\ ~Find~real~values~of~x$ ​

Answer 1

Solution -

p + q = x

Multiplying ( p - q ) to both sides of the equation

> (p - q)( p + q) = x ( p - q )

> [ p² - q² ] = x( p - q )

Now , p² = x - 1/x and q² = 1 - 1/x

So , p² - q²

> x - 1

> ( x - 1) = x( p - q )

Dividing both sides by x

> 1 - 1/x = p - q

So we get two equations

1. p + q = x

2. p - q = 1 - 1/x

Adding them

2p = x + 1 - 1/x

> 2p = ( x - 1/x ) + 1

> 2p = p² + 1

> ( p² - 2p + 1 ) = 0

Hence, p = 1

Now

p = ( x - 1/x )½

> [ x - 1/x ]^½ = 1

Squaring

> x - 1/x = 1

> x² - 1 = x

> x² - x - 1 = 0

The only real value of x possible Is the golden ratio ;

x = ( 1 + √5 ) /2 .

This is the required answer.

______________________________________

Answer 2

Question :

$\sf \: Given~that~p+q=x ~ \\\\ \sf \: where~p= \bigg(x-\frac{1}{x}\bigg)^{\frac{1}{2}}~and~q= \bigg(1-\frac{1}{x}\bigg)^{\frac{1}{2}}\\\\ \sf ~Find~real~values~of~x$

${\bigstar \: {\underline{\pmb{\sf{\red{\underline{Required \: \: solution}}}}}}}$

$\sf \: \implies \: P + Q = X$

Multiply ( p - q ) to both sides of the equation

• ( p - q ) ( p + q ) = x ( p - q )

• [ p² - q² ] = x ( p - q )

Now,

$\underline{ \boxed{ \sf{ \purple{p² \: = x \frac{ - 1}{x} \: and \: {q}^{2} = 1 \frac{ - 1}{x} }}}}$

So :

• p² - q²

• x - 1

• ( x - 1 )

• x ( p - q )

Divide both side of x :

$\underline{ \boxed{ \sf{ {1 \frac{ - 1}{x} = p - q}}}}$

So, we get two equations

$\sf \: 1) \: p + q = x$

$\sf \: 2) \: p - q = 1 \sf\frac{ - 1}{x}$

Adding :

$\underline{ \boxed{ \sf{ \purple{2p = x + 1 \frac{ - 1}{x} }}}}$

$\underline{ \boxed{ \sf{ \purple{2p = (x \frac{ - 1}{x} ) + 1}}}}$

$\underline{ \boxed{ \sf{ \purple{2p = {p}^{2} + 1 }}}}$

$\underline{ \boxed{ \sf{ \purple{( {p}^{2} - 2p + 1) = 0}}}}$

Hence, p = 1

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \huge \underline \red { \sf \: Now}$

$\underline{ \boxed{ \sf{ \pink{p = (x - \frac{ - 1}{x}) {}^{ \frac{1}{2} } }}}}$

$\underline{ \boxed{ \sf{ \pink{(x - \frac{1}{x} {)}^{ \frac{1}{2} } = 1 }}}}$

Squarting :

$\underline{ \boxed{ \sf{ {x \frac{ - 1}{x} = 1 }}}}$

$\underline{ \boxed{ \sf{ { {x}^{2} - 1 = x }}}}$

$\underline{ \boxed{ \sf{ { {x}^{2} - x - 1 = 0}}}}$

• The only real value of x possible is the golden ratio

$\underline{ \boxed{ \sf \red{ {x = \frac{(1 + \sqrt{5} )}{2} }}}}$

So finally your answer is done ;)