solve-for-x-R-12-4x-x-2-1-4-1-8x-2x-2-2x-2-8x-13-

Question Number 174427 by mnjuly1970 last updated on 31/Jul/22

s o l v e f o r x \in R :

\sqrt[4]{12 + 4 x^{} - x^{2}} + \sqrt{1 + 8 x - 2 x^{2}} = 2 x^{2} - 8 x + 13

Answered by MJS_new last updated on 02/Aug/22

\sqrt[4]{12 + 4 x - x^{2}} + \sqrt{1 + 8 x - 2 x^{2}} = 13 - 8 x + 2 x^{2}

x = \sqrt{y} + 2

\sqrt[4]{16 - y} + \sqrt{9 - 2 y} = 2 y + 5

now obviously y = 0 because 2 + 3 = 5 \Rightarrow

x = 2

Commented by mnjuly1970 last updated on 01/Aug/22

t h a n k s a l o t s i r

Commented by MJS_new last updated on 01/Aug/22

I {don}^{'} t think there are other solutions \in C

Sirs, you have to test your solutions with

the given equation!

Commented by MJS_new last updated on 02/Aug/22

there was a typo x = \sqrt{y} + 2

at least {it}^{'} s obvious to me . to prove y = 0 is

easy, simply insert . to prove {there}^{'} s no other

solution {you}^{'} ll have to solve it and show that

all the other solutions introduced by squaring

2 times are false . I {haven}^{'} t got the time .

Answered by behi834171 last updated on 01/Aug/22

1 + 8 x - 2 x^{2} = s^{2}, 12 + 4 x - x^{2} = t^{4}

\Rightarrow {\begin{cases} s + t = - s^{2} + 14 \Rightarrow t = 14 - s - s^{2} \\ s^{2} - 2 t^{4} = - 23 \end{cases}

\Rightarrow s^{2} - 2 {(14 - s - s^{2})}^{4} = - 23 \Rightarrow

s^{2} - 76832 + 10976 s + 10584 s^{2} - 784 s^{3} - 392 s^{4}

+ 10976 s - 1568 s^{2} - 1512 s^{3} + 112 s^{4} + 56 s^{5}

+ 10584 s^{2} - 1512 s^{3} - 1458 s^{4} + 108 s^{5} + 54 s^{6}

- 784 s^{3} + 112 s^{4} + 108 s^{5} - 8 s^{6} - 4 s^{7}

- 392 s^{4} + 56 s^{5} + 54 s^{6} - 4 s^{7} - 2 s^{8} = - 23

\Rightarrow 2 s^{8} + 8 s^{7} - 100 s^{6} - 324 s^{5} + 2018 s^{4}

+ 4592 s^{3} - 19691 s^{2} - 21952 s + 76809 = 0

{\begin{cases} \boldsymbol s_{1} = - 5.44 \Rightarrow - 2 \boldsymbol x^{2} + 8 \boldsymbol x + 1 = 29.6 \\ \boldsymbol s_{2} = - 3.36 \Rightarrow - 2 \boldsymbol x^{2} + 8 \boldsymbol x + 1 = 11.3 \end{cases}

\Rightarrow {\begin{cases} \boldsymbol x^{2} - 4 \boldsymbol x + 14.3 = 0 \Rightarrow \boldsymbol x = 2 \pm 3.2 \boldsymbol i \\ \boldsymbol x^{2} - 4 \boldsymbol x + 5.15 = 0 \Rightarrow \boldsymbol x = 2 \pm 1.07 \boldsymbol i \end{cases}

Commented by dragan91 last updated on 01/Aug/22

1 + 8 x - 2 x^{2} = s^{2}, 12 + 4 x - x^{2} = t^{4}

\Rightarrow {\begin{cases} ∣ s ∣ + ∣ t ∣= - s^{2} + 14 \Rightarrow t = 14 - s - s^{2} \\ s^{2} - 2 t^{4} = - 23 \end{cases}

{2 t}^{4} - s^{2} = 23

∣ t ∣= 14 - ∣ s ∣ - s^{2}

∣ s ∣= 3

∣ t ∣= 2

\Rightarrow 1 + 8 x - {2 x}^{2} = 9 \Rightarrow x = 2

\Rightarrow s^{2} - 2 {(14 - s - s^{2})}^{4} = - 23 \Rightarrow

s^{2} - 76832 + 10976 s + 10584 s^{2} - 784 s^{3} - 392 s^{4}

+ 10976 s - 1568 s^{2} - 1512 s^{3} + 112 s^{4} + 56 s^{5}

+ 10584 s^{2} - 1512 s^{3} - 1458 s^{4} + 108 s^{5} + 54 s^{6}

- 784 s^{3} + 112 s^{4} + 108 s^{5} - 8 s^{6} - 4 s^{7}

- 392 s^{4} + 56 s^{5} + 54 s^{6} - 4 s^{7} - 2 s^{8} = - 23

\Rightarrow 2 s^{8} + 8 s^{7} - 100 s^{6} - 324 s^{5} + 2018 s^{4}

+ 4592 s^{3} - 19691 s^{2} - 21952 s + 76809 = 0

{\begin{cases} \boldsymbol s_{1} = - 5.44 \Rightarrow - 2 \boldsymbol x^{2} + 8 \boldsymbol x + 1 = 29.6 \\ \boldsymbol s_{2} = - 3.36 \Rightarrow - 2 \boldsymbol x^{2} + 8 \boldsymbol x + 1 = 11.3 \end{cases}

\Rightarrow {\begin{cases} \boldsymbol x^{2} - 4 \boldsymbol x + 14.3 = 0 \Rightarrow \boldsymbol x = 2 \pm 3.2 \boldsymbol i \\ \boldsymbol x^{2} - 4 \boldsymbol x + 5.15 = 0 \Rightarrow \boldsymbol x = 2 \pm 1.07 \boldsymbol i \end{cases}

Commented by MJS_new last updated on 02/Aug/22

you must test these solutions, they are false

Commented by Tawa11 last updated on 02/Aug/22

Great sirs

Answered by dragan91 last updated on 03/Aug/22

\sqrt[4]{16 - 4 + 4 x - x^{2}} + \sqrt{9 - 8 + 8 x - {2 x}^{2}} = {2 x}^{2} - 8 x + 8 + 5

\sqrt[4]{2^{4} - {(x - 2)}^{2}} + \sqrt{3^{2} - 2 {(x - 2)}^{2}} = 2 {(x - 2)}^{2} + 5

\sqrt[4]{2^{4} - {(x - 2)}^{2}} ⩽ 2

\sqrt{3^{2} - 2 {(x - 2)}^{2}} ⩽ 3

2 {(x - 2)}^{2} ⩾ 5

only posible case 2 + 3 = 5

2 {(x - 2)}^{2} + 5 = 5

x - 2 = 0 \Rightarrow x = 2

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