Question

How do you solve the equation ${\left( {3x + 2} \right)^2} - 49 = 0$?

Answer 1

Hint: According to the given question, we have to solve the equation${\left( {3x + 2} \right)^2} - 49 = 0$.
So, first of all we have to write the constant term which is $49$ in terms of perfect square.
Now, we have to use the formula $\left( {{a^2} - {b^2}} \right)$ for factorisation of terms with the help of the formula which is mentioned below,
$ \Rightarrow \left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right).....................(A)$
Now, we have to put the both factorized terms equal to zero and find the values of $x$ in terms of numerical values.

Complete step by step solution:
Step 1: First of all we have to write the constant term which is $49$ in terms of perfect square,
$ \Rightarrow {\left( {3x + 2} \right)^2} - {\left( 7 \right)^2} = 0$
Step 2: Now, we have to use the formula (A) for $\left( {{a^2} - {b^2}} \right)$ for the expression which is obtained in the solution step 1with the help of the formula which is mentioned in the solution hint.
$ \Rightarrow \left( {3x + 2 + 7} \right)\left( {3x + 2 - 7} \right) = 0$
Now, we have solve the above expression by adding and subtracting the terms which can be added or subtracted,
$ \Rightarrow \left( {3x + 9} \right)\left( {3x - 5} \right) = 0$
Step 3: Now, we have to put the both factorized terms which are obtained in the solution step 2, equal to zero and find the values of $x$ in terms of numerical values.
$ \Rightarrow \left( {3x + 9} \right) = 0$ Or $\left( {3x - 5} \right) = 0$
Now, we have to solver the above expression and find the value of $x.$
\[
   \Rightarrow 3x = - 9 \\
   \Rightarrow x = \dfrac{{ - 9}}{3} \\
   \Rightarrow x = - 3 \\
 \]and\[
  3x = 5 \\
  x = \dfrac{5}{3} \\
 \]

Final solution: Hence, the solution for the equation ${\left( {3x + 2} \right)^2} - 49 = 0$ is \[x = \dfrac{5}{3}\] and \[x = - 3\].

Note:
It is necessary to write the constant term which is $49$ in terms of perfect square, so that we can easily use the formula of $\left( {{a^2} - {b^2}} \right)$ which is mentioned in the solution hint.
It is necessary to put the both factorized terms equal to zero and find the values of $x$ in terms of numerical values.