Question

Find the square root of following surd:
\[9 + 2\sqrt {14} \]

Answer 1

Hint: You can assume a surd with a variable and then evaluate its square and compare it with the given surd and solve the following equations and will get values of the surd which is your square root.

Complete step-by-step answer:
Let us suppose that $\sqrt {9 + 2\sqrt {14} } = \pm (\sqrt a + \sqrt b )$ ,where a, b are rational numbers
Now, squaring both sides, we get
\[ 9 + 2\sqrt {14} = {(\sqrt a + \sqrt b )^2} \\
  9 + 2\sqrt {14} = a + b + 2\sqrt {ab} \\
\]
Comparing both sides we get
$
  a + b = 9 \\
  \sqrt {ab} = \sqrt {14} {\text{ or }}ab = 14 \\
$
$b = \dfrac{{14}}{a}$
Now putting value of b in first expression, we get
$
   \Rightarrow a + \dfrac{{14}}{a} = 9 \\
   \Rightarrow {a^2} + 14 = 9a \\
   \Rightarrow {a^2} - 9a + 14 = 0 \\
 $
We can easily solve this quadratic equation using factorization. We can see 14=2 X 7 and 9=2+7 so using this we get
$ \Rightarrow {a^2} - 2a - 7a + 2 \times 7 = 0 \\
   \Rightarrow a(a - 2) - 7(a - 2) = 0 \\
   \Rightarrow (a - 2)(a - 7) = 0 \\
   \Rightarrow a = 2{\text{ or }}a = 7 \\
$
For $a = 2$ we have $b = \dfrac{{14}}{a} = \dfrac{{14}}{2} = 7$
and for $a = 7$ we have \[{\text{ }}b = \dfrac{{14}}{a} = \dfrac{{14}}{7} = 2\]
So our answer include \[{\text{(a,b) = (2,7) or (7,2) }}\] for which we get \[ \pm {\text{(}}\sqrt 2 {\text{ + }}\sqrt 7 {\text{) }}\] as our answer.

Note: You can also solve the quadratic equation by using formula $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ .Also there is a way to solve without quadratic equations, that is by observing the expression \[{\text{ }}a + b = 9;\sqrt {ab} = \sqrt {14} {\text{ or }}ab = 14\] You can deduce that a=2 and b=7 or b=7 and a=2.