<p><span style="font-size:20px;"><span style="color:#8e44ad;">Deductions from the Given Information</span></span></p>
<p>The given information provides us with a list of four numbers ($1$, $\sqrt{2}$, $x$, and $x^2$) and tells us that the range is $4$. So, we know the following:</p>
<p style="text-align: center;">$$the \: largest \: number - the \: smallest \: number = 4$$</p>
<p>But what's the largest number? What's the smallest number? If $x$ is less than $1$, say $\frac{1}{2}$, then $\sqrt{2}$ would be the largest number, $x^2 = \frac{1}{4}$ would be the smallest number, and the range would be $\sqrt{2} - \frac{1}{4}$. Clearly, this is <strong>NOT </strong>equal to $4$.</p>
<p>So $x$ must be greater than $1$ and the largest number must be $x^2$. That makes the smallest number $1$ and we can set up the following equation:</p>
<p style="text-align: center;">$$x^2 - 1 = 4$$ </p>
<p><span style="font-size:20px;"><span style="color:#27ae60;">Solving the Problem</span></span></p>
<p>The deduction above is really the hard part of this problem. Once it is made, it's rather trivial. We just solve for $x$ in the following way:</p>
<p>$$x^2 - 1 = 4$$</p>
<p>$$x^2 = 5$$</p>
<p>$$x=\sqrt{5} \approx 2.24$$</p>
<p>$2.24$ is definitely larger than $2$, so the correct answer is<strong><span style="color:#27ae60;"> A, Quantity A is larger</span></strong>. </p>