solve-x-x-x-1-x-x-2-x-1-x-2-2-

Question Number 137415 by mr W last updated on 02/Apr/21

s o l v e

x + \sqrt{x (x + 1)} + \sqrt{x (x + 2)} + \sqrt{(x + 1) (x + 2)} = 2

Commented by MJS_new last updated on 03/Apr/21

x_{1} = \frac{1}{24}

x_{2} \approx - 2.73908312241

x_{2} is a solution of x^{4} + x^{3} - 4 x^{2} + 2 x - \frac{1}{4} = 0

which I {couldn}^{'} t solve exactly and there are

no complex solutions .

Commented by liberty last updated on 03/Apr/21

h o w t o g e t x^{4} + x^{3} - 4 x^{2} + 2 x - \frac{1}{4} = 0 ?

Commented by MJS_new last updated on 03/Apr/21

\sqrt{A} + \sqrt{B} = D - \sqrt{C}

squaring, transforming

2 (\sqrt{A B} + D \sqrt{C}) = C + D^{2} - A - B

squaring, transforming

8 D \sqrt{A} \sqrt{B} \sqrt{C} = A^{2} + B^{2} + C^{2} + D^{4} - 2 (A B + A C + {A D}^{2} + B C + {B D}^{2} + {C D}^{2})

squaring, transforming, inserting

x^{5} + \frac{23}{24} x^{4} - \frac{97}{24} x^{3} + \frac{13}{6} x^{2} - \frac{1}{3} x + \frac{1}{96} = 0

trying factors of \frac{1}{96}

(x - \frac{1}{24}) (x^{4} + x^{3} - 4 x^{2} + 2 x - \frac{1}{4}) = 0

we can find 2 square factors

let x = z - \frac{1}{4}

z^{4} - \frac{35}{8} z^{2} + \frac{33}{8} z - \frac{259}{256} = 0

(z^{2} - a z - b) (z^{2} + a z - c) = 0

\Rightarrow

(1) a^{2} + b + c - \frac{35}{8} = 0

(2) a b - a c + \frac{33}{8} = 0

(3) b c + \frac{259}{256} = 0

(1) c = - a^{2} - b + \frac{35}{8}

(2) b = - \frac{a^{2}}{2} - \frac{33}{16 a} + \frac{35}{16} \Rightarrow c = - \frac{a^{2}}{2} + \frac{33}{16 a} + \frac{35}{16}

(3) \Rightarrow a^{6} - \frac{35}{4} a^{4} + \frac{371}{16} a^{2} - \frac{1089}{64} = 0

let a = \sqrt{r + \frac{35}{12}}

r^{3} - \frac{7}{3} r + \frac{107}{108} = 0

sadly this has no useable solution \dots

r_{1} = \frac{2 \sqrt{7}}{3} \sin \frac{\arcsin \frac{107 \sqrt{7}}{392}}{3}

r_{2} = \frac{2 \sqrt{7}}{3} \cos \frac{π + 2 \arcsin \frac{107 \sqrt{7}}{392}}{6}

r_{3} = - \frac{2 \sqrt{7}}{3} \sin \frac{π + \arcsin \frac{107 \sqrt{7}}{392}}{3}

\Rightarrow this makes no sense, {it}^{'} s “ {cheaper}^{″} to

approximately solve

x^{4} + x^{3} - 4 x^{2} + 2 x - \frac{1}{4} = 0

and check the solutions (squaring causes

false solutions)

x_{1} \approx - 2.73908 ★

x_{2} \approx . 200728 w r o n g

x_{3} \approx . 399135 w r o n g

x_{4} \approx 1.13922 w r o n g

Commented by mr W last updated on 03/Apr/21

t h a n k s s i r s!

Answered by mindispower last updated on 02/Apr/21

l e t Z = \sqrt{x (x + 1)} + \sqrt{x (x + 2)}

Y = x + \sqrt{(x + 1) (x + 2)}

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

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

Y^{2} - Z^{2} = 2 \dots 1

Y + Z = 2 \dots 2

(1) & (2) \Rightarrow Y - Z = 1

\Rightarrow Y = \frac{3}{2}

\Leftrightarrow (\frac{3}{2} - x) = \sqrt{(x + 1) (x + 2)} \Rightarrow - 3 x + \frac{9}{4} = 3 x + 2

\Rightarrow 6 x = \frac{1}{4} \Rightarrow x = \frac{1}{24} \dots .

\frac{1}{24} \dots i w i l l p o s t c o m p l e t s o l u t i o n l a t e r

i s o l v e d w i t h e \Rightarrow n o t \Leftrightarrow m u s t t c h e k i f \frac{1}{24} i s s o l u t i o n

Commented by mr W last updated on 02/Apr/21

t h a n k s s i r!

\frac{1}{24} i s a s o l u t o n . b u t i s i t t h e o n l y o n e ?

Answered by liberty last updated on 03/Apr/21

f o r x ⩾ 0

\sqrt{x} (\sqrt{x} + \sqrt{x + 1}) + \sqrt{x + 2} (\sqrt{x} + \sqrt{x + 1}) = 2

\sqrt{x} + \sqrt{x + 2} = \frac{2}{\sqrt{x} + \sqrt{x + 1}}

t h a t c a n b e w r i t t e n a s

\sqrt{x} + \sqrt{x + 2} = 2 \sqrt{x + 1} - 2 \sqrt{x}

3 \sqrt{x} + \sqrt{x + 2} = 2 \sqrt{x + 1}

10 x + 2 + 6 \sqrt{x (x + 2)} = 4 x + 4

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

9 x^{2} + 18 x = 9 x^{2} - 6 x + 1

\Leftrightarrow 24 x = 1; x = \frac{1}{24}

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