Question

How do you solve $ \sqrt {5x + 39} = x + 3 $ ?

Answer 1

Hint: First of all we will take the given expression and will use the concepts of squares and square-root. Also, apply that the square and square-root cancel each other during the solution. We will simplify the equations for the required resultant value.

Complete step-by-step answer:
Take the given equation:
 $ \sqrt {5x + 39} = x + 3 $
Apply square on both the sides of the equation-
 $ {\left( {\sqrt {5x + 39} } \right)^2} = {\left( {x + 3} \right)^2} $
Square and square cancel each other on the left hand side of the equation.
 $ \Rightarrow 5x + 39 = {\left( {x + 3} \right)^2} $
Now, we will use the whole square of two terms using the identity: $ {(a + b)^2} = {a^2} + 2ab + {b^2} $
 $ \Rightarrow 5x + 39 = {x^2} + 6x + 9 $
The above equation can be re-written as –
 $ \Rightarrow {x^2} + 6x + 9 = 5x + 39 $
Take all the terms on the left hand side of the equation. Remember when you move any terms from one side to another, the sign of the term also changes. Positive term changes to negative and vice-versa.
 $ \Rightarrow {x^2} + 6x + 9 - 5x - 39 = 0 $
Make a pair of like terms in the above equation.
 $ \Rightarrow {x^2} + \underline {6x - 5x} + \underline {9 - 39} = 0 $
Now simplify the above equation using the concepts that when there is one negative and one positive sign you have to do subtraction but sign of a bigger number.
 $ \Rightarrow {x^2} + x - 30 = 0 $
The above equation can be re-written as –
 $ \Rightarrow {x^2} + \underline {6x - 5x} - 30 = 0 $
Now make a pair of two-two terms. First two and last two terms.
 $ \Rightarrow \underline {{x^2} + 6x} - \underline {5x - 30} = 0 $
Take common factors from the above paired terms.
 $ \Rightarrow x(x + 6) - 5(x + 6) = 0 $
Take common factor
 $ \Rightarrow (x + 6)(x - 5) = 0 $
Therefore,
 $
   \Rightarrow x + 6 = 0 \\
   \Rightarrow x = ( - 6) \\
  $ or $
   \Rightarrow x - 5 = 0 \\
   \Rightarrow x = (5) \;
  $
So, the correct answer is “x=5 or x=-6”.

Note: Be careful in sign convention. When you move any term from one side to another, the sign also changes. Positive terms change to negative and negative term changes to positive. While doing simplification remember the golden rules-
Addition of two positive terms gives the positive term
Addition of one negative and positive term, you have to do subtraction and give sign of bigger numbers, whether positive or negative.
Addition of two negative numbers gives a negative number but in actual you have to add both the numbers and give a negative sign to the resultant answer.