Question

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

Answer 1

Hint: In this question, we have to find out the required expression from the given particulars.
We need to first understand the meaning of a radical symbol.
Radical:
The $\sqrt{} $ symbol that is used to denote square root of nth roots
Radical expression:
A radical expression is an expression containing a square root.
We need to convert the expression into a form which contains square root, after converting it we can find out the required solution.

Complete Step by Step Solution:
It is given that, \[{\left( {4x + 3} \right)^2} = 7\] .
We need to solve the equation and express \[{\left( {4x + 3} \right)^2} = 7\] in simplest radical form.
Since, Radical symbol is √ which is used to denote square root of nth roots and a radical expression is an expression containing a square root, we need to convert the given expression in a square root form.
Given that, \[{\left( {4x + 3} \right)^2} = 7\]
Taking square root both sides we get,
\[ \Rightarrow \left( {4x + 3} \right) = \pm \sqrt 7 \]
Again rearranging we get,
\[ \Rightarrow 4x = - 3 \pm \sqrt 7 \]
Solving we get,
\[ \Rightarrow x = \dfrac{{ - 3 \pm \sqrt 7 }}{4}\] , which is the required radical form.

\[\therefore x = \left( {n\pi + \dfrac{\pi }{6}} \right),\left( {n\pi + \dfrac{{5\pi }}{6}} \right)\] , for all integer value of \[n\] .

Note: Radical symbol:
A radical is a symbol that represents a particular root of a number. This symbol is shown below.
$\sqrt{}$.
Although this symbol looks similar to what is used in long division, a radical is different and has a vastly different meaning. The radical, by itself, signifies a square root. The square root of a number n is written as follows.
\[\sqrt n \] .
Radical expression:
A radical expression is an expression containing a square root.