Question

How do you solve the equation and express in simplest radical form for $ {\left( {4x + 3} \right)^2} = 7 $ ?

Answer 1

Hint:
According to the given question, we have to solve the equation and express in simplest radical form $ {\left( {4x + 3} \right)^2} = 7 $ .
So, first of all we have to convert the R.H.S term of the given expression as $ {\left( {4x + 3} \right)^2} = 7 $ in the form of whole square of the square root as mentioned below.
 $ \Rightarrow 7 = {\left( {\sqrt 7 } \right)^2} $
Now, we have to take the square root from both sides of the expression obtained from the sentence just mentioned above and we know that the square root of any real number has two values one is positive and other is negative as mentioned below.
 $ \Rightarrow \sqrt a = \pm a $ , where $ a $ is real number.
Now, we have to calculate the value of $ x $ for both positive sign and negative sign.

Complete step by step answer:
Step 1:
So, first of all we have to take square root from both sides of the given expression as $ {\left( {4x + 3} \right)^2} = 7 $ as mentioned in the solution hint.
 $ \Rightarrow {\left( {4x + 3} \right)^2} = {\left( {\sqrt 7 } \right)^2} $

Step 2:
Now, we have to take square root from both sides of the given expression as obtained in the solution step 1 and take both positive and negative sign of the real number as mentioned in the solution hint.
 $ \Rightarrow \left( {4x + 3} \right) = \pm \sqrt 7 $

Step 3:
Now, we have to take $ + \sqrt 7 $ and find the value of x.
 $
   \Rightarrow \left( {4x + 3} \right) = + \sqrt 7 \\
   \Rightarrow 4x = \sqrt 7 - 3 \\
   \Rightarrow x = \dfrac{{\sqrt 7 - 3}}{4} \\
  $

Step 4:
Now, we have to take $ - \sqrt 7 $ and find the value of x.
 $
   \Rightarrow \left( {4x + 3} \right) = - \sqrt 7 \\
   \Rightarrow 4x = - \sqrt 7 - 3 \\
   \Rightarrow x = \dfrac{{ - \sqrt 7 - 3}}{4} \\
  $

Step 5:
Now, we have to get two values of x from the solution step 3 and 4 as mentioned below.
 $ \Rightarrow x = \dfrac{{ - 3 \pm \sqrt 7 }}{4} $
Hence, the solution for the given expression as $ {\left( {4x + 3} \right)^2} = 7 $ in simple radical form is $ \dfrac{{ - 3 \pm \sqrt 7 }}{4} $ .

Note: It is necessary to convert the R.H.S term of the given expression as $ {\left( {4x + 3} \right)^2} = 7 $ in the form of whole square of the square root as mentioned in the solution hint.
It is necessary to take positive and negative signs of the R.H.S term taking the square root of both L.H.S and R.H.S sides.