x-3-2-x-4-x-5-6-x-2-2-Find-x-

Question Number 207206 by hardmath last updated on 09/May/24

\sqrt{x - 3 + 2 \sqrt{x - 4}} - \sqrt{x + 5 - 6 \sqrt{x - 2}} = 2

Find : x = ?

Answered by A5T last updated on 09/May/24

{(\sqrt{x - 4} + 1)}^{2} = x - 3 + 2 \sqrt{x - 4}

\Rightarrow \sqrt{x - 3 + 2 \sqrt{x - 4}} = \sqrt{x - 4} + 1

Q u e s t i o n \Rightarrow \sqrt{x - 4} - \sqrt{x + 5 - 6 \sqrt{x - 2}} = 1

\Rightarrow \sqrt{x - 4} = - 4 + 3 \sqrt{x - 2} \Rightarrow 3 \sqrt{x - 2} - \sqrt{x - 4} = 4

\Rightarrow u = \sqrt{x - 2}; v = \sqrt{x - 4}

\Rightarrow u^{2} - v^{2} = 2; 3 u - v = 4 \Rightarrow v = 3 u - 4

\Rightarrow u^{2} - {(3 u - 4)}^{2} = 2 \Rightarrow u^{2} - 9 u^{2} - 16 + 24 u = 2

\Rightarrow 8 u^{2} - 24 u + 18 = 0 \Rightarrow 4 u^{2} - 12 u + 9 = 0

\Rightarrow u = \frac{3}{2} \Rightarrow u^{2} + 2 = x = \frac{9}{4} + 2 = \frac{17}{4}

S u b s t i t u t i n g x = \frac{17}{4} g i v e s 1 \neq 2, c o n t r a d i c t i o n \to\leftarrow

H e n c e, n o s o l u t i o n e x i s t s .

Commented by hardmath last updated on 09/May/24

thank you dear professor

Commented by hardmath last updated on 10/May/24

thank you dear professor

Answered by Frix last updated on 09/May/24

1 . 0 ⩽ x - 4

\Rightarrow 4 ⩽ x \Rightarrow 1 ⩽ \sqrt{x - 3 + 2 \sqrt{x - 4}}

2 . 0 ⩽ x + 5 - 6 \sqrt{x - 2}

\Rightarrow 2 ⩽ x ⩽ 13 - 6 \sqrt{2} \lor x ⩾ 13 + 6 \sqrt{2}

=================

\Rightarrow 4 ⩽ x ⩽ 13 - 6 \sqrt{2} \lor x ⩾ 13 + 6 \sqrt{2}

A . 4 ⩽ x ⩽ 13 - 6 \sqrt{2}

\Rightarrow 1 + \sqrt{3} - \sqrt{6} ⩽ \sqrt{x - 3 + 2 \sqrt{x - 4}} - \sqrt{x + 5 - 6 \sqrt{x - 2}} ⩽ 1 - \sqrt{3} + \sqrt{6}

[\approx . 283 ⩽ lhs ⩽ 1.717]

B . 13 + 6 \sqrt{2} ⩽ x

\Rightarrow 4 < \sqrt{x - 3 + 2 \sqrt{x - 4}} - \sqrt{x + 5 - 6 \sqrt{x - 2}} ⩽ 1 + \sqrt{3} + \sqrt{6}

[\approx 4 < lhs ⩽ 5.182]

\Rightarrow no solution

Commented by hardmath last updated on 10/May/24

thank you dear professor