Why if I squared this equation $$|4x-3|=-|2x-1|$$ it will give a solution of $\frac{2}{3}$ and $1$?
But actually, it has no solution. Why squaring the equation will cause an error?
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Remember than squaring always introduces spurious roots. – Claude Leibovici Aug 05 '17 at 04:28
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Because by squaring we lose informations about the sign – Houssam Aug 05 '17 at 04:28
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https://math.stackexchange.com/questions/568780/why-cant-you-square-both-sides-of-an-equation – Hans Lundmark Aug 05 '17 at 06:58
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Because from your equation we have: $$4x-3=2x-1=0,$$ which is impossible.
Because $a=b\Leftrightarrow a^2=b^2$ is wrong.
The counterexample is your equation. Also, for example. $-1=1$ is wrong, but $(-1)^2=1^2$ is true.
Michael Rozenberg
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Try to substitute thsoe values inside to see what is going on
if you substitute $x=1$. you get $LHS=1$, $RHS=-1$, clearly $1 \neq -1$ but $1^2=(-1)^2$.
You can still use squaring to obtain possible solutions, however, you have to check whether they still satisfy the original problem.
Siong Thye Goh
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Squaring both sides and solving for $x$ is like solving $f(|4x-3|)=f(-|2x-1|)$ where $f(x)=x^2.$ This may yield an extraneous "solution" since $f(x)=x^2$ isn't one-to-one i.e. $$f(|4x-3|)=f(-|2x-1|) \nrightarrow |4x-3|=-|2x-1|.$$ If you seek to obtain an equation with the same solution set as the original, make sure you're working with one-to-one functions.