Determine all $a$ such that the following equation has exactly two solutions: $\dfrac{x}{5}+\dfrac{a^2}{x}=2$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$-\sqrt{5}$
Antwoord 2 correct
Fout
Antwoord 3 optie
For all $a$.
Antwoord 3 correct
Fout
Antwoord 4 optie
For no $a$.
Antwoord 4 correct
Fout
Antwoord 1 optie
The correct answer is not among the other options.
Antwoord 1 feedback
Correct: Note that the equation is not defined for $x=0$. Hence, for no $a$ is $x=0$ a solution.
If $a=0$: $\dfrac{x}{5}=2$ has one solution.
For $a\neq 0$ we have $\dfrac{x}{5}+\dfrac{a^2}{x}=2$.
Rewriting by multiplying by $x$ gives $\frac{1}{5}x^2-2x+a^2=0$.
$D=4-\frac{4}{5}a^2$.
$D=0$ for $a=-\sqrt{5}$ or $a=\sqrt{5}$.
$D>0$ for $-\sqrt{5}<a<\sqrt{5}$.
Hence, two solutions for $-\sqrt{5}<a<0$ or $0<a<\sqrt{5}$.
Go on.
Antwoord 2 feedback
Wrong: What is the equation for $a=0$?
Try again.
Antwoord 3 feedback
Wrong: Try again.
Antwoord 4 feedback
Wrong: Try again.