Question

The transformed equation of $ {x^4} + 8{x^3} + x - 5 = 0 $ by eliminating second term is:
A. $ {x^4} - 24{x^2} + 65x - 55 = 0 $
B. $ {x^4} + 24{x^2} - 65x + 55 = 0 $
C. $ {x^4} - 24{x^3} - 65x - 55 = 0 $
D. $ {x^4} + 24{x^2} + 65x + 55 = 0 $

Answer 1

Hint: Here, first compare given equation with standard equation and find coefficients of variable x with different order. Then apply the standard method used for transformation of the second term variable. Simplify the expression and replace the variable with x to get the result

Complete step-by-step answer:
We have given expression $ {x^4} + 8{x^3} + x - 5 = 0 $
Comparing above equation with standard equation we get the coefficients of variable as $ {a_1} = 1,{a_2} = 0,{a_3} = 1 $
In the given equation second term is $ {x^3} $
Second term can be eliminated using transformation x = y + h
And h = $ - \dfrac{{a_2}}{{4}} $
By putting values of coefficients we get h = − 2; x = y − 2
Now, we putting x = y − 2 in $ {x^4} + 8{x^3} + x - 5 = 0 $
We get \[{(y - 2)^4} + 8{(y - 2)^3} + (y - 2) - 5 = 0\]
On simplifying, we have
\[{y^4} - 8{y^3} + 24{y^2} - 32y + 16 + 8{y^3} - 48{y^2} - 96y - 64 + y - 2 - 5 = 0\]
Adding or subtracting all like terms
$\Rightarrow$ \[{y^4} - 24{y^2} + 65y - 55 = 0\]
Now replace y by x, we get
\[{x^4} - 24{x^2} + 65x - 55 = 0\]

So, the correct answer is “Option A”.

Note: In these types of questions, expand the expression obtained carefully and insure that the term asked to eliminate in question is not found in the result. In case the term is also in the final expression then check the simplifications step. Transformation of a function means we have changed the formula slightly and moved the graph around. We can replace the variable y by x or x by t it does not change the function. Also if options are given then simplify the function as per requirement. Always take care of the coefficients as if coefficient has negative sign then the coefficient should be taken as negative not the only numerical value.