Solve for real numbers the following system of equations { ((a(a+1) = b−1)),((a^2 (b+3)+2a = −1)) :}


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Question Number 146854 by mathdanisur last updated on 16/Jul/21
$Solve for real numbers the following system of equations { ((a(a+1) = b−1)),((a^2 (b+3)+2a = −1)) :}$
$S o l v e f o r r e a l n u m b e r s t h e f o l l o w i n g$ $s y s t e m o f e q u a t i o n s$ ${\begin{cases} a (a + 1) = b - 1 \\ a^{2} (b + 3) + 2 a = - 1 \end{cases}$
Commented by Mrsof last updated on 16/Jul/21

$a (a + b) + 4 = b + 3$ $a^{2} (a (a + 1) + 4) + 2 a = - 1$ $\Rightarrow a^{4} + a^{3} + 4 a^{2} + 2 a + 1 = 0$ $\Rightarrow a^{3} (a + 1) + 2 a^{2} + 2 a + 2 a^{2} + 1 = 0$ $\Rightarrow a^{3} (a + 1) + 2 a (a + 1) + (2 a^{2} + 1) = 0$ $\Rightarrow (a + 1) (a^{3} + 2 a) + (2 a^{2} + 1) = 0$ $h a s n o s o l u t i o n i n R$
Commented by mathdanisur last updated on 16/Jul/21

$t h a n k y o u S e r$
	Answered by liberty last updated on 16/Jul/21
	${ ((a(a+1)=b−1⇒b=a^2 +a+1)),((a^2 (b+3)+2a=−1)) :} ⇒a^2 (a^2 +a+4)+2a+1=0 ⇒a^4 +a^3 +4a^2 +2a+1=0 ⇒(a^2 +pa+1)(a^2 +qa+1)= a^4 +a^3 +4a^2 +2a+1 ⇒a^4 +(p+q)a^3 +(2+pq)a^2 +(p+q)a+1= a^4 +a^3 +4a^2 +2a+1 ⇒ { ((p+q=1)),((2+pq=4)),((p+q=2)) :} the equation inconsistent$
	${\begin{cases} a (a + 1) = b - 1 \Rightarrow b = a^{2} + a + 1 \\ a^{2} (b + 3) + 2 a = - 1 \end{cases}$ $\Rightarrow a^{2} (a^{2} + a + 4) + 2 a + 1 = 0$ $\Rightarrow a^{4} + a^{3} + 4 a^{2} + 2 a + 1 = 0$ $\Rightarrow (a^{2} + p a + 1) (a^{2} + q a + 1) =$ $a^{4} + a^{3} + 4 a^{2} + 2 a + 1$ $\Rightarrow a^{4} + (p + q) a^{3} + (2 + p q) a^{2} + (p + q) a + 1 =$ $a^{4} + a^{3} + 4 a^{2} + 2 a + 1$ $\Rightarrow {\begin{cases} p + q = 1 \\ 2 + p q = 4 \\ p + q = 2 \end{cases} t h e e q u a t i o n i n c o n s i s t e n t$
	Commented by mathdanisur last updated on 16/Jul/21

	$t h a n k y o u S e r$
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