Question

How do you simplify \[\sqrt{4{{x}^{2}}+9{{x}^{4}}+16}\] ?

Answer 1

Hint: The problems like these are very easy to understand and solve if we can get an in-depth understanding of the topic in general. We need to go through topics like quadratic equations and polynomials in order to solve problems efficiently. In our given problem we need to simplify it by, first of all observing the equation and then finding all possible perfect squares which are available inside the square root. After that, we can take the square root of all the square products (if at all possible), and then write it down in the form of a positive value and a negative value.

Complete step by step answer:
Now, we start off with the solution by writing,
We first try to write all the terms as perfect squares as,
\[\begin{align}
  & \sqrt{4{{x}^{2}}+9{{x}^{4}}+16} \\
 & =\sqrt{{{\left( 2x \right)}^{2}}+{{\left( 3{{x}^{2}} \right)}^{2}}+{{\left( 4 \right)}^{2}}} \\
\end{align}\]
We can further write this down in simpler ways as,
\[=\sqrt{{{\left( 3{{x}^{2}} \right)}^{2}}+2\cdot 3\cdot \dfrac{3}{2}{{x}^{2}}+{{\left( 4 \right)}^{2}}+{{\left( \dfrac{3}{2} \right)}^{2}}-{{\left( \dfrac{3}{2} \right)}^{2}}}\]
In the above step we add and subtract \[{{\left( \dfrac{3}{2} \right)}^{2}}\] , so that we can form at least one perfect square. Now, we write continuing from the above that,
\[=\sqrt{{{\left( 3{{x}^{2}}+\dfrac{3}{2} \right)}^{2}}+{{\left( 4 \right)}^{2}}-{{\left( \dfrac{3}{2} \right)}^{2}}}\]
We could only make one perfect square out of it. Now performing the required addition we get,
\[=\sqrt{{{\left( 3{{x}^{2}}+\dfrac{3}{2} \right)}^{2}}+\dfrac{55}{4}}\]
Now, here we can see that we cannot simplify the equation further, thus the above equation is our simplest form.

Note:
For such problems, we need to be very thorough with our concepts of quadratic equations and polynomial equations to efficiently solve this problem. We also need to observe the equation very carefully and find out all the possible squares in the given polynomial so that we can find the square root very easily and efficiently. We also need to be very careful while taking the square root, because it always has two signs, one positive and negative.