17-x-1-4-17-x-1-4-2-find-x-

Question Number 83608 by john santu last updated on 04/Mar/20

\sqrt[4]{17 + x} + \sqrt[4]{17 - x} = 2

find x

Commented by mr W last updated on 04/Mar/20

L H S ⩾ \sqrt[4]{2 \times 17} > \sqrt[4]{2 \times 16} > \sqrt[4]{16} = 2 = R H S

\Rightarrow n o s o l u t i o n f o r L H S = R H S

Commented by jagoll last updated on 04/Mar/20

yes . i got x = \emptyset

Commented by john santu last updated on 04/Mar/20

okay

Commented by MJS last updated on 04/Mar/20

x = a + b i

scetch it; let b > 0 [similar for b < 0]

z_{1} = 17 + a + b i lies above the real axis

z_{2} = 17 - a - b i lies under the real axis

the angles of z_{1} and z_{2} are only of \pm the same

values if a = 0 . but then ∣ z_{1}^{\frac{1}{4}} ∣=∣ z_{2}^{\frac{1}{4}} ∣⩾ \sqrt{17} > 2

otherwise the angles and the absolutes are

different and I {don}^{'} t see a chance the sum

of the roots might be real

Answered by behi83417@gmail.com last updated on 04/Mar/20

17 + x = m, 17 - x = n

\Rightarrow {\begin{cases} m + n = 34 \\ m^{4} + n^{4} = 2 \end{cases} \Rightarrow {\begin{cases} let : m + n = p, mn = q \\ \Rightarrow m^{4} + n^{4} = {[{(m + n)}^{2} - 2 mn]}^{2} - {2 m}^{2} n^{2} \\ \Rightarrow m^{4} + n^{4} = {(p^{2} - 2 q)}^{2} - {2 q}^{2} = p^{4} - {4 qp}^{2} + {2 q}^{2} \end{cases}

\Rightarrow {\begin{cases} p = 34 \\ p^{4} - {4 qp}^{2} + {2 q}^{2} = 2 \Rightarrow 34^{4} - 4 \times 34^{2} \times q + {2 q}^{2} = 2 \end{cases}

\Rightarrow q^{2} - 2312 q + 668167 = 0

\Rightarrow q = \frac{2312 \pm \sqrt{2312^{2} - 4 \times 668167}}{2} ⋍ 1973, 338

\Rightarrow {\begin{cases} p = 34 \\ q = 1973, 338 \end{cases}

\Rightarrow z^{2} - 34 z + (1973, 338) = 0 \Rightarrow z = {\begin{cases} z_{1, 2} = 34 \pm 82 i \\ z_{3, 4} = 34 \pm 14 i \end{cases}

\Rightarrow {\begin{cases} 17 + x = 34 \pm 82 i \Rightarrow x = 17 \pm 82 i \\ 17 - x = 34 \pm 82 i \Rightarrow x = - 17 \mp 82 i \\ 17 + x = 34 \pm 14 i \Rightarrow x = 17 \pm 14 i \\ 17 - x = 34 \pm 14 i \Rightarrow x = - 17 \mp 14 i \end{cases}

Commented by mr W last updated on 04/Mar/20

n i c e s o l u t i o n s i r!

Commented by MJS last updated on 04/Mar/20

none of these solve the given equation

Commented by MJS last updated on 04/Mar/20

I {don}^{'} t think this has any solution

Commented by behi83417@gmail.com last updated on 04/Mar/20

thank you dear masters : mrW sir and

MJS sir .

any way, i think this is a nice method

for solving such questions .

mybe someone likes to try this two :

1 . \sqrt[4]{97 - x} + \sqrt[4]{x} = 5

2 . \sqrt{1 - x^{2}} = {(a - \sqrt{x})}^{2}