Question

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Answer 1

5.

\(A=\dfrac{2\sqrt{x}}{x-2\sqrt{x}+4}\)

\(\Leftrightarrow Ax-2A\sqrt{x}+4A=2\sqrt{x}\)

\(\Leftrightarrow Ax-2A\sqrt{x}+4A=2\sqrt{x}\)

\(\Leftrightarrow Ax-2\left(A-1\right)\sqrt{x}+4A=0\)

\(\Delta'=A^2-2A+1-4A^2=-3A^2-2A+1\ge0\)

\(\Leftrightarrow\left(1-3A\right)\left(A+1\right)\ge0\)

\(\Leftrightarrow-1\le A\le\dfrac{1}{3}\)

\(A=0\left(\text{vì }A\ge0\right)\Leftrightarrow x=0\)

Answer 2

6.

\(P=\dfrac{4\sqrt{x}-13}{x-2\sqrt{x}+1}\)

\(\Leftrightarrow Px-2\left(P+2\right)\sqrt{x}+P+13=0\)

\(\Delta'=P^2+4P+4-P^2-13P=-9P+4\ge0\Leftrightarrow P\le\dfrac{4}{9}\)

\(P=0\Leftrightarrow4\sqrt{x}-13=0\Leftrightarrow x=\dfrac{169}{16}\)