- #1
yucheng
- 232
- 57
- Homework Statement
- Solve the following:
- Relevant Equations
- ##x^2-3|x|-2=0##
I used the identity ##\sqrt{x^2}=|x|## and completed the square as follows:
\begin{align*}
x^2-3|x|-2&=0 \tag1\\
\sqrt{x^4}-3\sqrt{x^2}-2&=0 \tag2\\
(\sqrt{x^2}-\frac{3}{2})^2-\frac{9}{4}-2&=0 \tag3\\
(\sqrt{x^2}-\frac{3}{2})^2&=\frac{17}{4} \tag4\\
\sqrt{x^2}-\frac{3}{2}&=\pm\frac{\sqrt{17}}{2} \tag5\\
\sqrt{x^2}&=\frac{3 \pm\sqrt{17}}{2} \tag6\\
x&= \pm\frac{3 \pm\sqrt{17}}{2} \tag7
\end{align*}
After substituting the solutions back into the equations, the solutions are:##x=±\frac{3+\sqrt{17}}{2}##
Which step did the extraneous solution ##x=±\frac{3−\sqrt{17}}{2}## originate? The method I used to search for this step is by checking from which step on did the extraneous solution satisfy the equation. I tried (5) but it was not. Or, was it because I used the identity ##\sqrt{x^2}=|x|##? It seems unlikely...
\begin{align*}
x^2-3|x|-2&=0 \tag1\\
\sqrt{x^4}-3\sqrt{x^2}-2&=0 \tag2\\
(\sqrt{x^2}-\frac{3}{2})^2-\frac{9}{4}-2&=0 \tag3\\
(\sqrt{x^2}-\frac{3}{2})^2&=\frac{17}{4} \tag4\\
\sqrt{x^2}-\frac{3}{2}&=\pm\frac{\sqrt{17}}{2} \tag5\\
\sqrt{x^2}&=\frac{3 \pm\sqrt{17}}{2} \tag6\\
x&= \pm\frac{3 \pm\sqrt{17}}{2} \tag7
\end{align*}
After substituting the solutions back into the equations, the solutions are:##x=±\frac{3+\sqrt{17}}{2}##
Which step did the extraneous solution ##x=±\frac{3−\sqrt{17}}{2}## originate? The method I used to search for this step is by checking from which step on did the extraneous solution satisfy the equation. I tried (5) but it was not. Or, was it because I used the identity ##\sqrt{x^2}=|x|##? It seems unlikely...
Last edited: