Question

If x be a real number such that $ {x^3} + 4x = 8 $ , then what is the value of the expression $ {x^7} + 64{x^2} $ ?
A) $ 124 $
B) $ 125 $
C) $ 128 $
D) $ 132 $

Answer 1

Hint: The given equation is a cubic equation with x as its variable. In order to find the value of the given expression $ {x^7} + 64{x^2} $ , we have to modify this expression in such a way that the value of the given equation can be substituted in it in order to eliminate the variable x.

Complete step-by-step answer:
Given to us are an equation $ {x^3} + 4x = 8 $ and an expression $ {x^7} + 64{x^2} $ whose value should be calculated.
From the given equation, we can find $ {x^3} $ by rearranging the terms. So this equation can be written as $ {x^3} = 8 - 4x $
In order to calculate the value of the expression $ {x^7} + 64{x^2} $ , we have to modify it. This can be done as follows.
 $ x \times {x^6} + 64{x^2} $
This expression can also be written as
 $ x \times {\left( {{x^3}} \right)^2} + 64{x^2} $
Now we can substitute the calculated value of $ {x^3} $ in this expression. So we can now write this expression as
 $ x \times {\left( {8 - 4x} \right)^2} + 64{x^2} $
Now we can use the formula
 $ {\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2} $ to expand $ {\left( {8 - 4x} \right)^2} $ from the above expression.
This gives $ x\left( {64 - 64x + 16{x^2}} \right) + 64{x^2} $
We can now eliminate the bracket and multiply the terms inside the bracket with the term x outside.
 $ 64x - 64{x^2} + 16{x^3} + 64{x^2} $
By solving this we get
 $ 64x + 16{x^3} = 16\left( {{x^3} + 4x} \right) $
The value of $ {x^3} + 4x $ is already given to us so we now substitute that value in the above expression to get $ 16\left( {{x^3} + 4x} \right) = 16\left( 8 \right) = 128 $
Therefore the value of the given expression $ {x^7} + 64{x^2} $ is $ 128 $ i.e. option C.
So, the correct answer is “Option C”.

Note: It is to be noted that in order to find the value of the given expression we should have the value of x but since we don’t have the value of x, we modify the given expression in order to eliminate the variable x from it using the given equation.