Question

Find three numbers if the cube of the first number exceeds their product by $ 2 $ , the cube of the second number is smaller than their product by $ 3, $ and the cube of the third number exceeds their product by $ 3. $

Answer 1

Hint: First convert the given word statements in the form of mathematical expressions. Find correlation among each mathematical expression and find unknowns accordingly.

Complete step-by-step answer:
Let us suppose that the three unknown numbers are - $ x,{\text{ y, and z}} $
Convert given three conditions in the form of mathematical expression -
First given condition is if the cube of the first number exceeds their product by $ 2 $
 $
  {x^3} - 2 = xyz \\
  \therefore {x^3} = xyz + 2\;{\text{ }}.......{\text{(1)}} \;
  $
Now, similarly given second condition is –
The cube of the second number is smaller than their product by $ 3, $
 $ {y^3} = xyz - 3\;{\text{ }}........{\text{(2)}} $
The cube of the third number exceeds their product by $ 3. $
 $ {z^3} = xyz + 3{\text{ }}.........{\text{(3)}} $
Multiplying equations $ (1),\;{\text{(2) and (3)}} $
We get,
 $ {x^3}{y^3}{z^3} = (xyz + 2)\underline {(xyz - 3)(xyz + 3)} $
As per the property of difference of the two squares –
 $
  (xyz - 3)(xyz + 3) = [{(xyz)^2} - {(3)^2}] \\
  (xyz - 3)(xyz + 3) = {x^2}{y^2}{z^2} - 9 \\
  \therefore {x^3}{y^3}{z^3} = (xyz + 2)({x^2}{y^2}{z^2} - 9) \\
  \therefore {x^3}{y^3}{z^3} = xyz({x^2}{y^2}{z^2} - 9) + 2({x^2}{y^2}{z^2} - 9) \\
  \therefore {x^3}{y^3}{z^3} = {x^3}{y^3}{z^3} - 9xyz + 2{x^2}{y^2}{z^2} - 18 \;
  $
Simplify the above equation – as like terms with same sign cancels each other
 $ $ So, $ {x^3}{y^3}{z^3}{\text{ from both the sides cancels each other}} $
 $
  0 = - 9xyz + 2{x^2}{y^2}{z^2} - 18 \\
  \therefore - 9xyz + 2{x^2}{y^2}{z^2} - 18 = 0 \\
  $
Arranging the above equation –
 $
  2{x^2}{y^2}{z^2} - 9xyz + - 18 = 0 \;
  \underline {2{x^2}{y^2}{z^2} - 12xyz} + \underline {3xyz - 18} = 0 \\
  2xyz(xyz - 6) + 3(xyz - 6) = 0 \\
  (xyz - 6)(2xyz + 3) = 0 \\
  xyz - 6 = 0{\text{ or 2xyz + 3 = 0}} \\
  {\text{xyz = 6 or xyz = }}\dfrac{{ - 3}}{2} \\
  $
Put, $ xyz = 6 $ in equation $ (1) $
 $
   \Rightarrow {x^3} = xyz + 2 \\
  \therefore {x^3} = 6 + 2 \\
  \therefore {x^3} = 8 \\
  \therefore x = 2 \;
  $
Similarly Put, $ xyz = 6 $ in equation $ (2) $
 $
  {y^3} = xyz - 3\; \\
  {y^3} = 6 - 3 \\
  {y^3} = 3 \\
  \sqrt[3]{{{y^3}}} = \sqrt[3]{3} \\
  y = \sqrt[3]{3}\;{\text{ }} \\
  {\text{y = }}{{\text{3}}^{1/3}} \;
  $ [Cube and cube roots cancel each other]
Now, put known value in equation $ (3) $
 $
\Rightarrow {z^3} = xyz + 3{\text{ }} \\
\Rightarrow {z^3} = 6 + 3 \\
\Rightarrow {z^3} = 9 \\
\Rightarrow \sqrt[3]{{{z^3}}} = \sqrt[3]{9}\;{\text{ (As 9 = }}{{\text{3}}^2}) \\
\Rightarrow z = {3^{2 \times 1/3}} \\
\Rightarrow z = {3^{2/3}} \;
  $
Repeat the same process for $ {\text{xyz = }}\dfrac{{ - 3}}{2} $ and place in the equations –
Put, $ xyz = \dfrac{{ - 3}}{2} $ in equation $ (1) $
\[
   \Rightarrow {x^3} = xyz + 2 \\
  \therefore {x^3} = \dfrac{{ - 3}}{2} + 2 \\
  \therefore {x^3} = \dfrac{{ - 3 + 4}}{2} \\
  \therefore {x^3} = \dfrac{1}{2} \;
 \]
Taking cube roots on both the sides, and apply property of cubes and cube roots cancel each other
 $ \therefore x = \dfrac{1}{{{2^{1/3}}}} $
Similarly Put, $ xyz = \dfrac{{ - 3}}{2} $ in equation $ (2) $
 $
\Rightarrow {y^3} = xyz - 3\; \\
\Rightarrow {y^3} = \dfrac{{ - 3}}{2} - 3 \\
\Rightarrow {y^3} = \dfrac{{ - 3 - 6}}{2} \\
\Rightarrow {y^3} = \dfrac{{ - 9}}{2} \\
\Rightarrow \sqrt[3]{{{y^3}}} = \sqrt[3]{{\dfrac{{ - 9}}{2}}} \\
\Rightarrow y = {(\dfrac{{ - 9}}{2})^{1/3}} \;
  $
Now, put known value in equation $ (3) $
 $
\Rightarrow {z^3} = \dfrac{{ - 3}}{2} + 3{\text{ }} \\
\Rightarrow {z^3} = \dfrac{{ - 3 + 6}}{2} \\
\Rightarrow {z^3} = \dfrac{3}{2} \\
\Rightarrow z = {(\dfrac{3}{2})^{1/3}} \;
  $
Thus, the required solution is –
When,
  $
\Rightarrow xyz = 6, \\
   \Rightarrow x = 2,\;y = {3^{1/3}},\;z = {3^{2/3}} \\
  xyz = \dfrac{{ - 3}}{2}, \\
   \Rightarrow x = \dfrac{1}{{{2^{1/3}}}},\;y = {(\dfrac{{ - 9}}{2})^{1/3}},\;z = {(\dfrac{3}{2})^{1/3}} \;
  $

Note: Always double check the conversion of the mathematical forms of the given conditions in the form of word statements. Find the correlation between the known and unknown terms and find solutions accordingly.