10. What is the value of $x$ in the equation $\frac{1+2 x+3 x^{2}}{3+2 x+x^{2}}=3$ ?
A -5
B -4
C -3
D -2
E -1
Solution
D

We note first that $3+2 x+x^{2}=2+(1+x)^{2}$. Therefore for all real numbers $x$, we have $3+2 x+x^{2} \geq 2$. Hence for all real numbers $x$, it follows that $3+2 x+x^{2} \neq 0$.
Therefore we can multiply both sides of the equation given in the question by $3+2 x+x^{2}$. In this way, we have
$$
\begin{aligned}
\frac{1+2 x+3 x^{2}}{3+2 x+x^{2}}=3 & \Leftrightarrow 1+2 x+3 x^{2}=3\left(3+2 x+x^{2}\right) \\
& \Leftrightarrow 1+2 x+3 x^{2}=9+6 x+3 x^{2} \\
& \Leftrightarrow 4 x+8=0 \\
& \Leftrightarrow x=-2 .
\end{aligned}
$$

FOR INVESTIGATION
10.1 Find all the solutions of the following equations.
(a) $\frac{1+3 x+5 x^{2}}{5+3 x+x^{2}}=1$.
(b) $\frac{5+7 x+2 x^{2}}{7+10 x+3 x^{2}}=1$.