Question

مطلوب است محاسبه xهای حقیقی: $$3(x+1) \sqrt{ x^{2} +x+3}-3 x^{2} -4x-7=0 $$ $$ \Longrightarrow 3(x+1) \sqrt{ x^{2} +x+3} - (x+1)^{2}-2( x^{2} +x+3)=0 $$

Answer 1

$$ \sqrt{ x^{2} +x+3} =a \wedge x+1=b \Longrightarrow 3ab- b^{2} -2 a^{2} =0 \Longrightarrow ab- b^{2} +2ab-2 a^{2} =0 \Longrightarrow b(a-b)-2a(a-b)=0 \Longrightarrow (a-b)(b-2a)=0 \Longrightarrow [a=b \Longrightarrow \sqrt{ x^{2} +x+3} =x+1 \Longrightarrow x+3=2x+1 \Longrightarrow x=2]\vee [b=2a \Longrightarrow x+1=2 \sqrt{ x^{2} +x+3} \Longrightarrow 3 x^{2} +2x+11=0 \Longrightarrow "x" not "real"]$$