Question

The difference between two numbers is 14 and the difference between their squares is 448, then find the numbers

Answer 1

Answer:

$\large{\underline{\boxed{\sf 23 \: \: and \: \: 9}}}$

Step-by-step explanation:

Let the two numbers be x and y respectively.

Given that ;

The difference between the two numbers is 14.

⇒ x - y = 14.... (i)

Difference between their squares is 448.

⇒ x² - y² = 448

⇒ (x + y)(x - y) = 448

⇒ (x + y) * 14 = 448

⇒ x + y = $\sf \dfrac{448}{14}$

⇒ x + y = 32 .....(ii)

_______________________

On adding equation (i) and (ii) -

⇒ x - y + x + y = 14 + 32

⇒ 2x = 46

⇒ x = $\sf \dfrac{46}{2}$

⇒ x = 23

Substituting the value of x in (i)

⇒ x - y = 14

⇒ 23 - y = 14

⇒ y = 23 - 14

⇒ y = 9

_______________________

Hence, the two required numbers are 23 and 9.

Answer 2

$\Huge{\boxed{Solution}}}$

===================

Given :

1. The difference between

  two numbers is 14

2. The difference between

   their squares is 448.

===================

To find :

Required number.

===================

Let us take :

Let the two number be

'x' and 'y'.

===================

Difference of between two numbers

⇒ x - y = 14

===================

Difference between their square

⇒ x² - y² = 448

Here, an identity is used

⇒ x²-y² = (x+y) (x-y)

⇒ (x+y) (x-y) = 448

⇒ (x+y) × 14 = 448  (Value of x-y is 14)

⇒ x+y = $\dfrac{448}{14}$

⇒ x+y = 32

Here we get the value

of x+y which is 32

===================

Now we have to add

x-y and x+y to find the value

of 'x' and we have

the value of x-y which is

14 and value of x+y which is

32.

Let us add

⇒ x + y + x - y = 14 + 32 ( y-y will be cancelled )

⇒ 2x = 14 + 32

⇒ 2x = 46

⇒ x = 46/2

⇒ x = 23

Here we get the value

of 'x' which is 23

===================

To find the second number

we have to put the value

of 'x' in (x - y = 14)

Let us put the

value of 'x' in

x - y = 14

⇒ x - y = 14

⇒ 23 - y = 14

⇒ 23 - 14 = y

⇒ 9 = y

Therefore value of y is 9

Here we get the value of 'y'

which is 9

===================

Therefore the two required

number is 23 and 9