Hello
I have an equation $\displaystyle \frac {\sqrt{x}-2}{(\sqrt{x} -1)^2} = 0$
If I try to solve this by multiplying both sides by same as denominator I get into complicated scenarios. Is there an easy way to handle this sort of thing?
Angus
Hello
I have an equation $\displaystyle \frac {\sqrt{x}-2}{(\sqrt{x} -1)^2} = 0$
If I try to solve this by multiplying both sides by same as denominator I get into complicated scenarios. Is there an easy way to handle this sort of thing?
Angus
Do you get complicated scenarios? If I multiply both sides by the denominator, I get:
$\displaystyle \sqrt{x}-2=0$
But I wouldn't even have to do that. If a fraction is equal to 0, then the numerator has to be 0. There is no other possibility. So by glancing, I would immediately be able to come to the above conclusion.
Basically it's because $\displaystyle 0$ divided by anything is $\displaystyle 0$.
Let's start with $\displaystyle 0$.
Do you understand that $\displaystyle 0=\frac{0}{2}$?
And that $\displaystyle \frac{0}{56}=0$?
So... $\displaystyle 0=\frac{0}{1}=\frac{0}{2}=\frac{0}{3}=\frac{0}{4}= ...$
If, as a completely random example, $\displaystyle \frac{x+7}{3}=0...$
...and, of course, $\displaystyle 0=\frac{0}{3}$
Then $\displaystyle \frac{x+7}{3}=0=\frac{0}{3}$
Can you see here that this means that the numerators are equal?
With fractions, the only possible way you can get a result of 0 is if the numerator of the fraction is $\displaystyle 0$.
Absolutely ! Another way to think about it:
$\displaystyle \frac{x+7}{3}=0$
Multiply both sides by $\displaystyle 3$ (the denominator):
$\displaystyle \frac{3(x+7)}{3}=0\times~3$
$\displaystyle x+7=0$
Again, the denominator's value is lost.
So in your original problem, we just have to glance once at the seemingly difficult equation and find that it's actually just asking you to solve:
$\displaystyle \sqrt{x}-2=0$