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Question Number 59861 by ANTARES VY last updated on 15/May/19

\sqrt{a + b \sqrt{c}} = \sqrt{\frac{a + \sqrt{a^{2} - b^{2} c}}{2}} + \sqrt{\frac{a - \sqrt{a^{2} - b^{2} c}}{2}} .

p r o v e

Commented by Kunal12588 last updated on 15/May/19

Commented by Kunal12588 last updated on 15/May/19

j u s t p u t “ b^{2} c^{″} i n p l a c e o f “ b^{″}

Answered by tanmay last updated on 15/May/19

\frac{a + \sqrt{a^{2} - b^{2} c}}{2}

= \frac{2 a + 2 \sqrt{(a + b \sqrt{c}) (a - b \sqrt{c})}}{4}

= \frac{a + b \sqrt{c} + a - b \sqrt{c} + 2 \sqrt{(a + b \sqrt{c) (a - b \sqrt{c)}}}}{4}

= \frac{{(\sqrt{a + b \sqrt{c}})}^{2} + {(\sqrt{a - b \sqrt{c}})}^{2} + 2 \sqrt{(a + b \sqrt{c}) (a - b \sqrt{c)}}}{4}

= \frac{{[\sqrt{a + b \sqrt{c}} + \sqrt{a - b \sqrt{c}}]}^{2}}{2^{2}}

s o

\sqrt{\frac{a + \sqrt{a^{2} - b^{2} c}}{2}} + \sqrt{\frac{a - \sqrt{a^{2} - b^{2} c}}{2}}

= \frac{\sqrt{a + b \sqrt{c}} + \sqrt{a - b \sqrt{c}}}{2} + \frac{\sqrt{a + b \sqrt{c}} - \sqrt{a - b \sqrt{c}}}{2}

= \sqrt{a + b \sqrt{c}}

Answered by Kunal12588 last updated on 15/May/19

l e t \sqrt{a + b \sqrt{c}} = \sqrt{m} + \sqrt{n} (1)

\Rightarrow a + \sqrt{b^{2} c} = m + n + 2 \sqrt{m n}

\Rightarrow m + n = a, 4 m n = b^{2} c

\Rightarrow m - n = \sqrt{{(m + n)}^{2} - 4 m n} = \sqrt{a^{2} - b^{2} c}

\Rightarrow m = \frac{a + \sqrt{a^{2} - b^{2} c}}{2}, n = \frac{a - \sqrt{a^{2} - b^{2} c}}{2}

f r o m (1)

\sqrt{a + b \sqrt{c}} = \sqrt{m} + \sqrt{n}

\Rightarrow \sqrt{a + b \sqrt{c}} = \sqrt{\frac{a + \sqrt{a^{2} - b^{2} c}}{2}} + \sqrt{\frac{a - \sqrt{a^{2} - b^{2} c}}{2}}

p r o v e d