95(1205). Solve the equation

$$
x^{2}+\frac{4 x^{2}}{(x+2)^{2}}=5
$$
Solution. I s p o s b. We have:

$$
x^{2}+\left(\frac{2 x}{x+2}\right)^{2}=5
$$

Introduce a new unknown $\frac{2 x}{x+2}=y$. Then we have the system:

$$
\left\{\begin{array}{l}
x^{2}+y^{2}=5 \\
x y=2(x-y)
\end{array}\right.
$$

Multiplying both sides of the second equation by -2 and adding it to the first, we get the equation

$$
(x-y)^{2}+4(x-y)-5=0
$$

from which $x-y=-5$ or $x-y=1$.

To find $x$ and $y$, we have two systems:

$$
\left\{\begin{array} { l } 
{ x - y = - 5 } \\
{ x y = - 1 0 }
\end{array} \quad \left\{\begin{array}{l}
x-y=1 \\
x y=2
\end{array}\right.\right.
$$

The first system has no real solutions, the second has two solutions: $x_{1}=-1, y_{1}=-2 ; x_{2}=2, y_{2}=1$. Therefore, the given equation has two solutions: $x_{1}=-1, x_{2}=2$.

II s p o o b.

$$
\begin{gathered}
(x+2)^{2} x^{2}+4 x^{2}=5(x+2)^{2},(x+2)^{2} x^{2}-4(x+2)^{2}+4 x^{2}-(x+2)^{2}=0 \\
(x+2)^{2}\left(x^{2}-4\right)+\left(2 x^{2}\right)-(x+2)^{2}=0 \\
(x+2)^{3}(x-2)+(3 x+2)(x-2)=0 \\
(x-2)\left((x+2)^{3}+3 x+2\right)=0, x_{1}=2 \\
(x+2)^{3}+3 x+2=0, x^{3}+6 x^{2}+12 x+8+3 x+2=0 \\
\left(x^{3}+1\right)+6 x(x+1)+9(x+1)=0, \quad(x+1)\left(x^{2}-x+1+6 x+9\right)=0 \\
x+1=0, x_{2}=-1
\end{gathered}
$$

The equation $x^{2}+5 x+10=0$ has no real roots. Answer: $-1 ; 2$.

$96(1206)$. Construct the graph of the function:
a) $y=\frac{\left(x^{2}-5 x+6\right)(1-x)}{x-3}$
b) $y=\frac{x^{4}-13 x^{2}+36}{4-x^{2}}$

Solution.
a) $y=\frac{\left(x^{2}-5 x+6\right)(1-x)}{x-3}=\frac{(x-2)(x-3)(1-x)}{x-3}=(x-2) \times$ $\times(1-x)=-x^{2}+3 x-2$ for $x \neq 3$.

The graph of the given function is the graph of the quadratic polynomial $y=-x^{2}+3 x-2=-\left(x-\frac{3}{2}\right)^{2}+\frac{1}{4}$ with a "punched out" point at $x=3$ (Fig. 43).

b) $y=\frac{x^{1}-13 x^{2}+36}{4-x^{2}}=\frac{\left(x^{2}-9\right)\left(x^{2}-4\right)}{4-x^{2}}=9-x^{2}$ for $x^{2} \neq 4$, i.e., for $|x| \neq 2$.

208

![](https://cdn.mathpix.com/cropped/2024_05_21_98932225530737b82071g-212.jpg?height=708&width=534&top_left_y=251&top_left_x=407)

Fig. 43

![](https://cdn.mathpix.com/cropped/2024_05_21_98932225530737b82071g-212.jpg?height=708&width=540&top_left_y=251&top_left_x=1129)

Fig. 44

The graph of the given function is the parabola $y=9-x^{2}$ with "punched out" points at $x=2$ and $x=-2$ (Fig. 44).