solve for x ∈ R : ((12+4x^ −x^( 2) ))^(1/4) +(√(1+8x−2x^( 2) )) = 2x^( 2) −8x +13 ■


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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
	$1+8x−2x^2 =s^2 ,12+4x−x^2 =t^4 ⇒ { ((s+t=−s^2 +14⇒t=14−s−s^2 )),((s^2 −2t^4 =−23)) :} ⇒s^2 −2(14−s−s^2 )^4 =−23⇒ s^2 −76832+10976s+10584s^2 −784s^3 −392s^4 +10976s−1568s^2 −1512s^3 +112s^4 +56s^5 +10584s^2 −1512s^3 −1458s^4 +108s^5 +54s^6 −784s^3 +112s^4 +108s^5 −8s^6 −4s^7 −392s^4 +56s^5 +54s^6 −4s^7 −2s^8 =−23 ⇒2s^8 +8s^7 −100s^6 −324s^5 +2018s^4 +4592s^3 −19691s^2 −21952s+76809=0 { ((s_1 =−5.44⇒−2x^2 +8x+1=29.6)),((s_2 =−3.36⇒−2x^2 +8x+1=11.3)) :} ⇒ { ((x^2 −4x+14.3=0⇒x=2±3.2i)),((x^2 −4x+5.15=0⇒x=2±1.07i)) :}$
	$1 + 8 x - 2 x^{2} = s^{2}, 12 + 4 x - x^{2} = t^{4}$ $\Rightarrow {\begin{cases} s + t = - s^{2} + 14 \Rightarrow t = 14 - s - s^{2} \\ s^{2} - 2 t^{4} = - 23 \end{cases}$ $\Rightarrow s^{2} - 2 {(14 - s - s^{2})}^{4} = - 23 \Rightarrow$ $s^{2} - 76832 + 10976 s + 10584 s^{2} - 784 s^{3} - 392 s^{4}$ $+ 10976 s - 1568 s^{2} - 1512 s^{3} + 112 s^{4} + 56 s^{5}$ $+ 10584 s^{2} - 1512 s^{3} - 1458 s^{4} + 108 s^{5} + 54 s^{6}$ $- 784 s^{3} + 112 s^{4} + 108 s^{5} - 8 s^{6} - 4 s^{7}$ $- 392 s^{4} + 56 s^{5} + 54 s^{6} - 4 s^{7} - 2 s^{8} = - 23$ $\Rightarrow 2 s^{8} + 8 s^{7} - 100 s^{6} - 324 s^{5} + 2018 s^{4}$ $+ 4592 s^{3} - 19691 s^{2} - 21952 s + 76809 = 0$ ${\begin{cases} s_{1} = - 5.44 \Rightarrow - 2 x^{2} + 8 x + 1 = 29.6 \\ s_{2} = - 3.36 \Rightarrow - 2 x^{2} + 8 x + 1 = 11.3 \end{cases}$ $\Rightarrow {\begin{cases} x^{2} - 4 x + 14.3 = 0 \Rightarrow x = 2 \pm 3.2 i \\ x^{2} - 4 x + 5.15 = 0 \Rightarrow x = 2 \pm 1.07 i \end{cases}$
	Commented by dragan91 last updated on 01/Aug/22
	$1+8x−2x^2 =s^2 ,12+4x−x^2 =t^4 ⇒ { ((∣s∣+∣t∣=−s^2 +14⇒t=14−s−s^2 )),((s^2 −2t^4 =−23)) :} 2t^4 −s^2 =23 ∣t∣=14−∣s∣−s^2 ∣s∣=3 ∣t∣=2 ⇒1+8x−2x^2 =9⇒x=2 ⇒s^2 −2(14−s−s^2 )^4 =−23⇒ s^2 −76832+10976s+10584s^2 −784s^3 −392s^4 +10976s−1568s^2 −1512s^3 +112s^4 +56s^5 +10584s^2 −1512s^3 −1458s^4 +108s^5 +54s^6 −784s^3 +112s^4 +108s^5 −8s^6 −4s^7 −392s^4 +56s^5 +54s^6 −4s^7 −2s^8 =−23 ⇒2s^8 +8s^7 −100s^6 −324s^5 +2018s^4 +4592s^3 −19691s^2 −21952s+76809=0 { ((s_1 =−5.44⇒−2x^2 +8x+1=29.6)),((s_2 =−3.36⇒−2x^2 +8x+1=11.3)) :} ⇒ { ((x^2 −4x+14.3=0⇒x=2±3.2i)),((x^2 −4x+5.15=0⇒x=2±1.07i)) :}$
	$1 + 8 x - 2 x^{2} = s^{2}, 12 + 4 x - x^{2} = t^{4}$ $\Rightarrow {\begin{cases} ∣ s ∣ + ∣ t ∣= - s^{2} + 14 \Rightarrow t = 14 - s - s^{2} \\ s^{2} - 2 t^{4} = - 23 \end{cases}$ ${2 t}^{4} - s^{2} = 23$ $∣ t ∣= 14 - ∣ s ∣ - s^{2}$ $∣ s ∣= 3$ $∣ t ∣= 2$ $\Rightarrow 1 + 8 x - {2 x}^{2} = 9 \Rightarrow x = 2$ $\Rightarrow s^{2} - 2 {(14 - s - s^{2})}^{4} = - 23 \Rightarrow$ $s^{2} - 76832 + 10976 s + 10584 s^{2} - 784 s^{3} - 392 s^{4}$ $+ 10976 s - 1568 s^{2} - 1512 s^{3} + 112 s^{4} + 56 s^{5}$ $+ 10584 s^{2} - 1512 s^{3} - 1458 s^{4} + 108 s^{5} + 54 s^{6}$ $- 784 s^{3} + 112 s^{4} + 108 s^{5} - 8 s^{6} - 4 s^{7}$ $- 392 s^{4} + 56 s^{5} + 54 s^{6} - 4 s^{7} - 2 s^{8} = - 23$ $\Rightarrow 2 s^{8} + 8 s^{7} - 100 s^{6} - 324 s^{5} + 2018 s^{4}$ $+ 4592 s^{3} - 19691 s^{2} - 21952 s + 76809 = 0$ ${\begin{cases} s_{1} = - 5.44 \Rightarrow - 2 x^{2} + 8 x + 1 = 29.6 \\ s_{2} = - 3.36 \Rightarrow - 2 x^{2} + 8 x + 1 = 11.3 \end{cases}$ $\Rightarrow {\begin{cases} x^{2} - 4 x + 14.3 = 0 \Rightarrow x = 2 \pm 3.2 i \\ x^{2} - 4 x + 5.15 = 0 \Rightarrow x = 2 \pm 1.07 i \end{cases}$
	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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