Question

How do you solve $\sqrt {x + 8} - \sqrt {x - 4} = 2$?

Answer 1

Hint: In this question first, we will rearrange the given expression and square the values and simply the expression to get the value of $x$. Finally we get the required answer.

Complete step-by-step solution:
We have the given expression as: $\sqrt {x + 8} - \sqrt {x - 4} = 2$
Now on transferring the term $\sqrt {x - 4} $ to the right-hand side, we get:
$ \Rightarrow \sqrt {x + 8} = 2 + \sqrt {x - 4} $
Now on squaring both the sides, we can write the expression as:
$ \Rightarrow {\left( {\sqrt {x + 8} } \right)^2} = {\left( {2 + \sqrt {x - 4} } \right)^2}$
Now we will apply the formula ${(a + b)^2} = {a^2} + 2ab + {b^2}$ on the right-hand side, the expression can be written as:
$ \Rightarrow {\left( {\sqrt {x + 8} } \right)^2} = \left( {{2^2} + 2 \times 2 \times \sqrt {x - 4} + {{\left( {\sqrt {x - 4} } \right)}^2}} \right)$
On simplifying, we get:
\[ \Rightarrow x + 8 = 4 + 4\sqrt {x - 4} + x - 4\]
Now on cancelling the terms in the left-hand side, we get:
\[ \Rightarrow x + 8 = 4\sqrt {x - 4} + x\]
On transferring $x$ from the right-hand side to the left-hand side, we get:
\[ \Rightarrow x - x + 8 = 4\sqrt {x - 4} \]
On simplifying the left-hand side, we get:
\[ \Rightarrow 8 = 4\sqrt {x - 4} \]
Now on transferring $4$ from the right-hand side to the left-hand side, we get:
\[ \Rightarrow \dfrac{8}{4} = \sqrt {x - 4} \]
On simplifying the left-hand side, we get:
\[ \Rightarrow 2 = \sqrt {x - 4} \]
Now on squaring the expression, we can write it as:
\[ \Rightarrow {2^2} = {\left( {\sqrt {x - 4} } \right)^2}\]
On simplifying, we get:
\[ \Rightarrow 4 = x - 4\]
On rearranging the terms, we can write the expression as:
$ \Rightarrow x = 4 + 4$
On simplifying, we get:
$x = 8$, which is the required solution.

Therefore the value of x is equal to 8.

Note: Now to check whether a solution is correct, we will put the value of $x = 8$ in the original expression to cross-check.
On substituting $x = 8$ in the left-hand side, we get:
$ \Rightarrow \sqrt {8 + 8} - \sqrt {8 - 4} $
On simplifying, we get:
$ \Rightarrow \sqrt {16} - \sqrt 4 $
On taking the square root, we get:
$ \Rightarrow 4 - 2$
On simplifying, we get:
$ \Rightarrow 2$, which is right-hand side, therefore the solution is correct.
It is to be remembered that when a term which is in addition or subtraction is transferred across the $ = $sign, its sign changes. Same happens with a term which is in multiplication or division.
It is to be remembered that the square of a square root value gets us the original number, which means ${\left( {\sqrt a } \right)^2} = a$.