Question

The equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+15-8\sqrt{x-1}}=2$ has:
1) No real root
2) At least one real root
3) Exactly one root
4) An infinite number of real roots

Answer 1

Hint: Here in this question we have been asked to evaluate the roots of the given equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+15-8\sqrt{x-1}}=2$ . For simplifying this expression we can use the following simplifications: $x+3-4\sqrt{x-1}={{\left( \sqrt{x-1}-2 \right)}^{2}}$ and $x+15-8\sqrt{x-1}={{\left( \sqrt{x-1}-4 \right)}^{2}}$ .

Complete step-by-step solution:
Now considering from the question we have been asked to evaluate the roots of the given equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+15-8\sqrt{x-1}}=2$ .
For simplifying this expression we can use the following simplifications: $x+3-4\sqrt{x-1}={{\left( \sqrt{x-1}-2 \right)}^{2}}$ and $x+15-8\sqrt{x-1}={{\left( \sqrt{x-1}-4 \right)}^{2}}$ .
By using these simplifications we will have $\Rightarrow \pm \left( \sqrt{x-1}-2 \right)\pm \left( \sqrt{x-1}-4 \right)=2$.
Now here 4 cases are possible let us evaluate them one by one.
Case 1: $ \left( \sqrt{x-1}-2 \right)+\left( \sqrt{x-1}-4 \right)=2$ .
By simplifying this further we will have
$\begin{align}
  & \Rightarrow 2\sqrt{x-1}-6=2 \\
 & \Rightarrow \sqrt{x-1}=4 \\
 & \Rightarrow x-1=16 \\
 & \Rightarrow x=17 \\
\end{align}$ .
In this case, the equation has only one real root.
Case 2: $ \left( \sqrt{x-1}-2 \right)-\left( \sqrt{x-1}-4 \right)=2$ .
By simplifying this further we can conclude that all real values of $x$ satisfying the condition $x\ge 1$ are roots for this equation.
Case 3: $ -\left( \sqrt{x-1}-2 \right)+\left( \sqrt{x-1}-4 \right)=2$ .
By simplifying this further we can conclude that the equation has no real roots in this case.
Case 4: $ -\left( \sqrt{x-1}-2 \right)-\left( \sqrt{x-1}-4 \right)=2$ .
By simplifying this further we will have
$\begin{align}
  & \Rightarrow -2\sqrt{x-1}+6=2 \\
 & \Rightarrow \sqrt{x-1}=2 \\
 & \Rightarrow x-1=4 \\
 & \Rightarrow x=5 \\
\end{align}$ .
In this case, the equation has only one real root.
Therefore we can conclude that the given equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+15-8\sqrt{x-1}}=2$ has an infinite number of real roots.
Hence we will mark the option “D” as correct.

Note: In the process of answering questions of this type we should be sure that we don’t even miss out any one of the possible conditions. Because missing any one condition can also lead us to end up having a wrong answer. The only way to solve questions of this type is to practice more questions.