Question

The difference of the squares of two natural numbers is 84. the square of the larger number is 25 times the smaller number. find the numbers.

Answer 1

Answer:

100 and 4.

Step-by-step explanation:

Let the two natural numbers be x and y

No we are given tha tThe difference of the squares of two natural numbers is 84.

$\Rightarrow x^2-y^2=84$ ---A

We are also given that the square of the larger number is 25 times the smaller number

$x^2=25y$ ---B

Substitute the value of $x^2$ in A

$\Rightarrow 25y-y^2=84$

$\Rightarrow y^2-25y+84=0$

$\Rightarrow y^2-21y-4y+84=0$

$\Rightarrow y(y-21)-4(y-21)=0$

$\Rightarrow (y-21)(y-4)=0$

$\Rightarrowy=21,4$

Substitute the values of y in B to get values of x

$x^2=25y$

At y = 21

$x^2=25(21)$

$x^2=525$

$x=\sqrt{525}$

$x=\sqrt{525}$

Since x is not natural number .So, we will reject this value

At y = 4

$x^2=25(4)$

$x^2=100$

$x=\sqrt{100}$

$x=10$

Thus the two natural numbers are 100 and 4.

Thsu

Answer 2

Answer:

Required two natural numbers are 10,4

Step-by-step explanation:

Let m and n are two natural numbers.(x>y)

According to the problem given,

The difference of the squares of two natural numbers is 84.

x²-y² = 84 ---(1)

The square of the larger number is 25 times the smaller number

x² = 25y ----(2)

/* put x²=25y in equation (1), we get

25y-y²=84

=> 0=y²-25y+84

=> y²-25y+84=0

/* Splitting the middle term,we get

=> y²-4y-21y+84=0

=> y(y-4)-21(y-4)=0

=> (y-4)(y-21)=0

=> y-4 = 0 Or y-21=0

=> y = 4 Or y = 21

Now , put y values in equation (2), we get

Case 1:

If y=4 then x² = 25×4

=> x = √(25×4)

=> x = 5×2

=> x = 10

case 2:

If y=21 then x² = 25×21

=> x = √(25×21) is not a natural number.

Therefore,

Required two natural numbers are 10,4

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