Question

The difference between 2 numbers is 2 their product is 80 the numbers

Answer 1

Given :

• The difference between 2 numbers is 2.
• Product of the numbers is 80.

To Find :

• The number

Solution :

Let the greater number be x.

Let the smaller number be y.

Case 1 :

Difference between greater number and smaller number is 2.

$\sf{x-y=2}$

$\sf{x=y+2}$ ___(1)

Case 2 :

Product of the greater and smaller number is 80.

Equation :

$\green{\implies}$ $\sf{xy=80}$

From (1), x = y + 2

$\green{\implies}$ $\sf{(y+2)y=80}$

$\green{\implies}$ $\sf{y^2\:+\:2y=80}$

$\green{\implies}$ $\sf{y^2\:+2y-80=0}$

$\green{\implies}$ $\sf{y^2+10y-8y-80=0}$

$\green{\implies}$ $\sf{y(y+10)-8(y+10)=0}$

$\green{\implies}$ $\sf{(y+10)\:\:(y-8)\:=0}$

$\green{\implies}$ $\sf{y+10=0\:\:or\:\:y-8=0}$

$\green{\implies}$ $\sf{y=-10\:\:or\:\:y=8}$

We have two cases here,

• When, y = - 10
• When, y = 8

Case 1 :

Substitute, y = - 10 in equation (1),

$\green{\implies}$ $\sf{x=y+2}$

$\green{\implies}$ $\sf{x=(-10)+2}$

$\green{\implies}$ $\sf{x=-10+2}$

$\green{\implies}$ $\sf{x\:=-8}$

$\large{\boxed{\sf{\red{Greater\:Number\:=\:x\:=\:-8}}}}$

$\large{\boxed{\sf{\red{Smaller\:Number\:=\:y\:=\:-10}}}}$

Case 2 :

Substitute, y = 8 in equation (1),

$\green{\implies}$ $\sf{x=y+2}$

$\green{\implies}$ $\sf{x=8+2}$

$\green{\implies}$ $\sf{x=10}$

$\large{\boxed{\sf{\purple{Greater\:Number\:=\:x\:=\:10}}}}$

$\large{\boxed{\sf{\purple{Smaller\:Number\:=\:y\:=\:8}}}}$

Answer 2

$\huge\underline\frak\red{To\:find}$

The number

$\huge\underline\frak\red{Solution}$

$\textbf{According to question}$

$\textbf{Let the numbers be x and y}$

$\implies\sf (x-y)=2$ ------(i)

$\implies\sf xy=80$

$\implies\sf x=\frac{80}{y}$

Putting the value of x in equation (i)

$\implies\sf x-y=2$

$\implies\sf \frac{80}{y}-y=2$

$\implies\sf \frac{80-y^2}{y}=2$

$\implies\sf y^2-2y-80=0$

$\implies\sf y^2+10y-8y-80=0$

$\implies\sf y(y+10)-8(y+10)=0$

$\implies\sf (y+10)(y-8)=0$

In case : 1

y = -10

or

In case : 2

y = 8

Putting the value y = -10 in equation (i)

In case : 1

$\implies\sf x-y=2$

$\implies\sf x-(-10)=2$

$\implies\sf x+10=2$

$\implies\sf x=-10+2=-8$

putting the value y = 8 in equation (i)

In case : 2

$\implies\sf x-y=2$

$\implies\sf x-8=2$

$\implies\sf x=8$

Result

Number in case : 1

x = -8 and y = -10

Number in case : 2

x = 8 and y = 8