Question

Solve $ \sqrt {3x + 1} - \sqrt {x - 1} = 2 $ ?

Answer 1

Hint: The value of x in the given algebraic equation can be found by using the method of transposition and squaring both the sides. Method of transposition involves doing the exact same mathematical thing on both sides of an equation with the aim of simplification in mind. This method can be used to solve various algebraic equations like the one given in question with ease.

Complete step-by-step answer:
In the given problem, we are required to solve for the value of x in the given algebraic expression.
Consider, $ \sqrt {3x + 1} - \sqrt {x - 1} = 2 $
Transposing $ \sqrt {x + 1} $ to right side of the equation, we get,
 $ \sqrt {3x + 1} = 2 + \sqrt {x - 1} $
Squaring both sides to remove the square roots, we get,
\[{\left( {\sqrt {3x + 1} } \right)^2} = {\left( {2 + \sqrt {x - 1} } \right)^2}\]
Opening the squares of the brackets,
 $ \left( {3x + 1} \right) = 4 + (x - 1) + 4\sqrt {x - 1} $
Again transposing the terms without square root to left side of the equation in order to isolate the square root,
 $ \left( {3x + 1} \right) - 4 - (x - 1) = 4\sqrt {x - 1} $
Adding, subtracting and cancelling the like terms, we get,
 $ 2x - 2 = 4\sqrt {x - 1} $
Taking out $ 2 $ common from left side of the equation, we get,
 $ 2(x - 1) = 4\sqrt {x - 1} $
 $ (x - 1) = 2\sqrt {x - 1} $
Now, we square both sides of the equation,
 $ {(x - 1)^2} = {\left( {2\sqrt {x - 1} } \right)^2} $
Opening the squares of the brackets, we get,
 $ {x^2} - 2x + 1 = 4(x - 1) $
Rearranging the like terms together,
 $ {x^2} - 6x + 5 = 0 $
Now, we have to factorize the quadratic equation obtained in order to find the values of x.
So, $ {x^2} - 6x + 5 = 0 $
 $ \Rightarrow (x - 5)(x - 1) = 0 $
Either $ (x - 5) = 0 $ or $ (x - 1) = 0 $
Either $ x = 5 $ or $ x = 1 $
Hence, the solution for the given algebraic equation is: $ x = 5 $ and $ x = 1 $
So, the correct answer is “ $ x = 5 $ and $ x = 1 $ ”.

Note: We have to be careful while transposing terms from one side of the equation to another and keep in mind to change the signs of the terms while doing so. There is no fixed way of solving a given algebraic equation. Algebraic equations can be solved in various ways. Linear equations in one variable can be solved by the transposition method with ease.