if:x+(√y)=a , (√x)+y=b find x,y


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Question Number 118386 by mohammad17 last updated on 17/Oct/20

$i f : x + \sqrt{y} = a, \sqrt{x} + y = b f i n d x, y$
	Answered by benjo_mathlover last updated on 17/Oct/20
	$let (√y) = t ∧ (√x) = q ⇒ { ((q^2 +t=a ...(×b))),((q+t^2 =b ...(×a))) :} ⇒ { ((bq^2 +bt=ab)),((aq+at^2 =ab)) :} ⇒bq^2 −aq+bt−at^2 =0 ⇒bq^2 −aq −(at^2 −bt)=0 ⇒b(q^2 −(a/b)q)−a(t^2 −(b/a)t)=0 ⇒b{(q−(a/(2b)))^2 −(a^2 /(4b^2 ))}−a{(t−(b/(2a)))^2 −(b^2 /(4a^2 ))} = 0$
	$l e t \sqrt{y} = t \land \sqrt{x} = q$ $\Rightarrow {\begin{cases} q^{2} + t = a . . . (\times b) \\ q + t^{2} = b . . . (\times a) \end{cases}$ $\Rightarrow {\begin{cases} {b q}^{2} + b t = a b \\ a q + {a t}^{2} = a b \end{cases}$ $\Rightarrow {b q}^{2} - a q + b t - {a t}^{2} = 0$ $\Rightarrow {b q}^{2} - a q - ({a t}^{2} - b t) = 0$ $\Rightarrow b (q^{2} - \frac{a}{b} q) - a (t^{2} - \frac{b}{a} t) = 0$ $\Rightarrow b {{(q - \frac{a}{2 b})}^{2} - \frac{a^{2}}{4 b^{2}}} - a {{(t - \frac{b}{2 a})}^{2} - \frac{b^{2}}{4 a^{2}}} = 0$
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