Question

Solve: $162{x^4} - 50$

Answer 1

Hint: First of all take the given expression and take out the common multiple from the expression and frame the equation in the form of the difference of two squares and use the identity for the difference of two squares and simplify for the required expression.

Complete step-by-step answer:
Take the given expression: $162{x^4} - 50$
Take out the common multiples common from the above expression –
$ = 2(81{x^4} - 25)$
The above expression can be written as the square of the terms –
$ = 2[{(9{x^2})^2} - {(5)^2}]$
Use the identity for the difference of two squares which can be expressed as- ${a^2} - {b^2} = (a - b)(a + b)$in the above expression.
$ = 2[(9{x^2} - 5)(9{x^2} + 5)]$
Again, the above expression can be re-written in the form of squares. Remember $\sqrt n \times \sqrt n = n$
\[ = 2[({(3x)^2} - {(\sqrt 5 )^2})(9{x^2} + 5)]\]
Apply the difference of two squares identity –
\[ = 2[((3x - \sqrt 5 )(3x + \sqrt 5 )(9{x^2} + 5)]\]
Hence, the required solution can be given by \[162{x^4} - 50 = 2[((3x - \sqrt 5 )(3x + \sqrt 5 )(9{x^2} + 5)]\]

Note: Remember the difference between the squares of the numbers and square root. Square of any number is always positive and the square root of any number can be plus or minus. Remember the identity for the difference of the terms and apply accordingly since it is the first and basic important step for the correct solution.