x-2-816-64-77-x-write-x-like-a-c-b-d-

Question Number 8351 by Nayon last updated on 09/Oct/16

x^{2} = 816 - 64 \sqrt{77}

x = ? [w r i t e x l i k e a \sqrt{c} + b \sqrt{d]}

Commented by ridwan balatif last updated on 09/Oct/16

x = \pm \sqrt{816 - 2 \times 32 \sqrt{77}}

x = \pm \sqrt{816 - 2 \sqrt{32^{2} \times 77}}

x = \pm \sqrt{816 - 2 \sqrt{78848}}

x = \pm \sqrt{(704 + 112) - 2 \sqrt{704 \times 112}}

x = \pm (\sqrt{704} - \sqrt{112})

x = \pm (8 \sqrt{11} - 4 \sqrt{7})

x = \pm 4 (2 \sqrt{11} - \sqrt{7})

Answered by Rasheed Soomro last updated on 09/Oct/16

Let {(a \sqrt{c} + b \sqrt{d})}^{2} = 816 - 64 \sqrt{77} a, b, c, d \in N

{(a \sqrt{c})}^{2} + 2 (a \sqrt{c}) (b \sqrt{d}) + {(b \sqrt{d})}^{2} = 816 - 64 \sqrt{77}

a^{2} c + 2 a b \sqrt{c d} + b^{2} d = 816 - 64 \sqrt{77}

C o m p a r i n g

a^{2} c + b^{2} d = 816 \land 2 ab = 64 \land cd = 77

cd = 77 = 7 \times 11 \Rightarrow L et c = 7, d = 11

a^{2} c + b^{2} d = 816 \Rightarrow 7 a^{2} + 11 b^{2} = 816

2 a b = 64 \Rightarrow b = 32 / a

7 a^{2} + 11 b^{2} = 816 \Rightarrow 7 a^{2} + 11 {(32 / a)}^{2} = 816

7 a^{2} + 11 (1024 / a^{2}) = 816

7 a^{2} + 11264 / a^{2} = 816

7 a^{4} - 816 a^{2} + 11264 = 0

a^{2} = \frac{- (- 816) \pm \sqrt{{(- 816)}^{2} - 4 (7) (11264)}}{2 (7)}

T a k i n g + o u t o f \pm we w i l l g e t a f r a c t i o n a l

n u m b e r, w h i c h i s a g a i n s t o u r a s s u m p t i o n .

H e n c e w e w i l l t a k e - s i g n o u t o f \pm .

a^{2} = \frac{816 - \sqrt{{(- 816)}^{2} - 4 (7) (11264)}}{14} = \frac{816 - 592}{14} = 16

a = \pm 4 \Rightarrow b = 32 / a = 32 / \pm 4 = \pm 8

a \sqrt{c} + b \sqrt{d} = (\pm 4) \sqrt{7} \pm 8 \sqrt{11} = \pm 4 (\sqrt{7} + 2 \sqrt{11})