Question

How do you solve \[\sqrt {8x} + 1 = 65\] and check the solution?

Answer 1

Hint: In the given problem we need to solve this for ‘x’. We can solve this using the transposition method. The common transposition method is to do the same thing (mathematically) to both sides of the equation, with the aim of bringing like terms together and isolating the variable (or the unknown quantity). That is we group the ‘x’ terms one side and constants on the other side of the equation.

Complete step by step solution:
Given, \[\sqrt {8x} + 1 = 65\] .
We transpose \[1\] which is present in the left hand side of the equation to the right hand side of the equation by subtracting ‘1’ on the right hand side of the equation.
 \[\sqrt {8x} = 65 - 1\]
 \[\sqrt {8x} = 64\]
Now squaring on both sides we have,
 \[{\left( {\sqrt {8x} } \right)^2} = {\left( {64} \right)^2}\]
We know that square and square root will cancels out
 \[8x = {\left( {64} \right)^2}\]
 \[8x = {\left( {8 \times 8} \right)^2}\]
 \[8x = 8 \times 8 \times 8 \times 8\]
We transpose \[8\] to the right hand side of the equation by dividing \[8\] on the right hand side of the equation.
 \[x = \dfrac{{8 \times 8 \times 8 \times 8}}{8}\]
 \[x = 8 \times 8 \times 8\]
 \[ \Rightarrow x = 512\] is the required answer.
We can check whether the obtained solution is correct or wrong. All we need to do is substituting the value of ‘x’ in the given problem.
 \[\sqrt {8\left( {512} \right)} + 1 = 65\]
 \[\sqrt {4096} + 1 = 65\]
We know that 4096 is a perfect square
 \[
  64 + 1 = 65 \\
   \Rightarrow 65 = 65 \;
 \]
Hence the obtained answer is correct.
So, the correct answer is “ x = 512”.

Note: In the above we did the transpose of addition and subtraction. Similarly if we have multiplication we use division to transpose. If we have division we use multiplication to transpose. Follow the same procedure for these kinds of problems.