Question

the difference between two numbers is 5 and their product is 14 then find the difference between their cubes ​

Answer 1

solution has been explained in the image

Hope it helps you

Answer 2

Given :

• Difference between two number is 5.
• The product is 14.

To Find :

• Difference between their cubes.

Solution :

Let the greater number be x.

Let the smaller number be y.

Case 1 :

Difference between x and y is 5.

Equation :

$\longrightarrow$ $\sf{x-y=5}$

$\sf{x=y+5\:\:\:(i)}$

Case 2 :

Product of x and y gives us 14.

Equation :

$\longrightarrow$ $\sf{xy=14}$

From (i), x = y + 5,

$\longrightarrow$ $\sf{(y+5)y=14}$

$\longrightarrow$ $\sf{y^2+5y=14}$

$\longrightarrow$ $\sf{y^2+5y-14=0}$

Now, we can clearly see that the equation formed is a quadratic equation.

Let's use the factorization method to solve for y.

$\longrightarrow$ $\sf{y^2\:+\:7y-2y-14=0}$

$\longrightarrow$ $\sf{y(y+7)-2(y+7)=0}$

$\longrightarrow$ $\sf{(y+7)\:\:(y-2)\:\:=0}$

$\longrightarrow$ $\sf{y+7=0\:\:or\:\:y-2=0}$

$\longrightarrow$ $\sf{y=-7\:\:or\:\:y=2}$

We have two values of y, one as -7 and the other one as 2.

So let's consider both the cases.

Case 1, when y = - 7 :

Substitute, y = - 7 in equation (i),

$\longrightarrow$ $\sf{x=y+5}$

$\longrightarrow$ $\sf{x=-7+5}$

$\longrightarrow$ $\sf{x=-2}$

•°• Greater number, x = - 2

Smaller Number, y = - 7.

Difference between cubes :

$\longrightarrow$ $\sf{x^3-y^3}$

$\longrightarrow$ $\sf{(-2)^3\:-\:(-7)^3}$

$\longrightarrow$ $\sf{(-8)-(-343)}$

$\longrightarrow$$\sf{-8+343}$

$\longrightarrow$ $\sf{335}$

$\large{\boxed{\sf{\purple{Difference\:between\:number\:=\:335}}}}$

Case 2, when y = 2,

Substitute, y = 2, in equation (i),

$\longrightarrow$ $\sf{x=y+5}$

$\longrightarrow$ $\sf{x=2+5}$

$\longrightarrow$ $\sf{x=7}$

•°• Greater Number, x = 7

Smaller Number, y = 2.

Difference between cubes, when y = 2,

$\longrightarrow$ $\sf{(x)^3\:-\:(y)^3}$

$\longrightarrow$ $\sf{(7)^3-(2)^3}$

$\longrightarrow$ $\sf{343-8}$

$\longrightarrow$ $\sf{335}$

So, we see that the difference between the cubes of number when one number is negative or positive is always the same in both cases. So from here onwards, if you come across such question, instead of considering both case, consider only one case. Both cases gives the same result, so better save your time.