Question

solving equation (see the question) following roots are obtain .
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Answer 1

Answer:

$\boxed{\boxed {\boxed { \bold{ \huge{ \pink{x = - 4p \: or \: 3p}}}}}}$

Step-by-step explanation:

$\dfrac{x + \sqrt{12p - x} }{x - \sqrt{12p - x} } = \dfrac{ \sqrt{p} + 1 }{ \sqrt{p} - 1} \\ \pink{ \bold{applying \: componendo \: and \: dividendo}} \\ \dfrac{x + \sqrt{12p - x} + x - \sqrt{12p - x } }{x + \sqrt{12p - x } - x + \sqrt{12p - x} } = \dfrac{ \sqrt{p} + 1 + \sqrt{p} - 1 }{ \sqrt{p} + 1 - \sqrt{p} + 1} \\ \dfrac{2x}{2 \sqrt{12p - x} } = \dfrac{2 \sqrt{p} }{2} \\ \dfrac{x}{ \sqrt{12p - x } } = \sqrt{p} \\ x = \sqrt{p} \sqrt{12p - x} \\ \bold{ \blue{squaring \: both \: sides}} \\ {x}^{2} = p \: (12p - x) \\ {x}^{2} = 12 {p}^{2} - px \\ {x}^{2} + px - 12 {p}^{2} = 0 \\ {x}^{2} + (4 - 3)px - 12 {p}^{2} = 0 \\ {x}^{2} + 4px - 3px - 12 {p}^{2} = 0 \\ x(x + 4 p) - 3p(x + 4p) = 0 \\ (x + 4p) \: (x - 3p) = 0 \\ if \\ \: x + 4p = 0 \\ x = - 4p \\ if \\ x - 3p = 0 \\ x = 3p$