Question

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Determine the range of
(x² + x + 1)/(x² - x + 1)​

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Answer 1

Answer:

Let y=x2+x+1x2−x+1

⇒yx2+yx+y=x2−x+1

⇒(x−1)x2+(y+1)x+(y−1)=0

If x∈R, then 

Discriminant ≥0

⇒(y+1)2−4(y−1)2≥0

⇒−3y2+10y−3≥0

⇒3y2−10y+3≤0

⇒(3y−1)(y−3)≤0

⇒31≤y≤3

∴ Range =[31,3]

Answer 2

$\begin{gathered}\huge\blue{\mid{\fbox{\tt{QUESTION}}\mid}} \\ \end{gathered}$

(x² + x + 1)/(x² - x + 1)

$\begin{gathered}\huge\blue{\mid{\fbox{\tt{SOLUTION}}\mid}} \\ \end{gathered}$

Let y=x2+x+1x2−x+1

⇒yx2+yx+y=x2−x+1

⇒(x−1)x2+(y+1)x+(y−1)=0

If x∈R, then 

Discriminant ≥0

⇒(y+1)2−4(y−1)2≥0

⇒−3y2+10y−3≥0

⇒3y2−10y+3≤0

⇒(3y−1)(y−3)≤0

⇒31≤y≤3

∴ Range =[31,3]