# How do I find the solutions of $|x-2|^{10x^2-1}=|x-2|^{3x}$?

How do I find the solutions of the following equation: $$|x-2|^{10x^2-1}=|x-2|^{3x}\ ?$$

I found that this equation has 5 solutions, 4 positive and 1 negative by looking the graph:

Question: How do I compute the values of these roots manually?

We see that $$x=2$$ is one solution. Let $$x\ne 2$$.

Taking $$\log$$ we get $$(10x^2-1)\log|x-2|=3x\log|x-2|$$

So one solution is $$\log |x-2| = 0\implies |x-2| =1 \implies x-2=\pm1$$, so $$x=3$$ or $$x=1$$.

Say $$\log |x-2| \ne 0$$ then $$10x^2-1 = 3x$$ so $$x= {1\over 2}$$ and $$x=-{1\over 5}$$.

So rearranging gives $$|x-2|^{10x^2-1}-|x-2|^{3x}=0$$ $$|x-2|^{3x}(|x-2|^{10x^2-3x-1}-1)=0$$ So either $$x=2$$ to achieve zero in the first factor, $$|x-2|=1\implies x=1,3$$ in order for the second factor to be $$1-1=0$$. We can also have $$10x^2-3x-1=0\implies x=-\frac15 , \frac12$$ where the power in the second factor is $$0$$ and hence also causes $$1-1=0$$.

Hint

Either $$x=2$$or$$|x-2|^{10x^2-3x-1}=1$$what are all the answers of $$a^b=1$$? (In our case, $$x=3$$ is one answer. What about the others?)

We get easy that $$x=2$$ is one solution. Now let $$x\neq 2$$, then it must be $$10x^2-1=3x$$. Can you finish? Hint: $$x=3$$ and $$x=1$$ are also solutions.

• Yes thank you sir. – Namami Shanker Mar 27 at 18:45
• This does not give all of the solutions. – Peter Foreman Mar 27 at 18:47
• The point about $x=3$ and $x=1$ is that these make $|x-2|=1$, and then $|x-2|^p = 1$ for any $p$. If $x \ne 1, 2, 3$, then we must have $10 x^2-1 = 3x$, because $a^t$ is a one-to-one function of $t$ if $0 < a < 1$ or $a > 1$. – Robert Israel Mar 27 at 18:54