Giải thích các bước giải:
\(
\begin{array}{l}
P = \frac{{x^2 + 2x}}{{2x + 10}} + \frac{{x - 5}}{x} + \frac{{50 - 5x}}{{2x(x + 5)}} \\
a)Đkxđ:\left\{ {\begin{array}{*{20}c}
{2x + 10 \ne 0} \\
{x \ne 0} \\
{2x(x + 5) \ne 0} \\
\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}c}
{x \ne 0} \\
{x \ne - 5} \\
\end{array}} \right. \\
b)P = \frac{{x^2 + 2x}}{{2x + 10}} + \frac{{x + 5}}{x} + \frac{{50 - 5x}}{{2x(x + 5)}}(x \ne 0;x \ne - 5) \\
= \frac{{x^2 + 2x}}{{2(x + 5)}} + \frac{{x + 5}}{x} + \frac{{50 - 5x}}{{2x(x + 5)}} \\
= \frac{{x(x^2 + 2x) + 2(x + 5)^2 + 50 - 5x}}{{2x(x + 5)}} \\
= \frac{{x^3 + 2x^2 + 2x^2 + 20x + 50 + 50 - 5x}}{{2x(x + 5)}} \\
= \frac{{x^3 + 4x^2 + 15x + 100}}{{2x(x + 5)}} \\
= \frac{{(x + 5)(x^2 - x + 20)}}{{2x(x + 5)}} \\
= \frac{{x^2 - x + 20}}{{2x}} \\
c)P = 0 \Rightarrow \frac{{x^2 - x + 20}}{{2x}} = 0(x \ne 0;x \ne - 5) \\
\Leftrightarrow x^2 - x + 20 = 0 \\
\end{array}
\)
=> Pt vô nghiệm
+)\(
\begin{array}{l}
P = \frac{1}{4} \Leftrightarrow \frac{{x^2 - x + 20}}{{2x}} = \frac{1}{4}(x \ne 0;x \ne - 5) \\
\Leftrightarrow 4x^2 - 4x + 80 = 2x \\
\Leftrightarrow 4x^2 - 6x + 80 = 0 \\
\end{array}
\)
=>Pt vô nghiệm
Vậy không có giá trị của x để P=0 và P=1/4
d)\(
\begin{array}{l}
d)P > 0 \Rightarrow \frac{{x^2 - x + 20}}{{2x}} > 0 \\
\Rightarrow 2x > 0(do:x^2 - x + 20 > 0) \\
\Leftrightarrow x > 0 \\
\end{array}
\)
=>\(
P < 0 \Leftrightarrow x < 0;x \ne - 5
\)
