Question

Two positive numbers differ by 5 and the sum of their squares is 193 . find the two numbers

Answer 1

Answer:

Let the two positive whole numbers be x and y.

Given that these numbers differ by = 5

Therefore x - y = 5 

x = 5 + y

Also given that the sum of their squares = 193

Therefore

x² + y² = 193

Substituting the value of x

(5 + y)² + y² = 193

25 + y² + 10y + y ² = 193

2y² + 10y - 168 = 0

y² + 5y - 84 = 0

y² + 12y - 7y - 84 = 0

y(y + 12) - 7(y + 12)

(y + 12) (y - 7) = 0

y = -12 or y = 7

Rejecting y = -12 as given they are positive whole numbers.

Therefore y = 7

x - 7 = 5

x = 12

The two numbers are 12 and 7.

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Answer 2

The two numbers are 7 and 12.

Step-by-step explanation:

Let x and y be the two numbers.

Then it is given that x = y + 5 ------------(1)

x² + y² = 193 -------------(2)

Substituting (1) in (2), we get:

(y + 5)² + y² = 193

25 + y² + 10y + y ² = 193

2y² + 10y - 168 = 0

Let's solve the quadratic equation.

y² + 5y - 84 = 0

y² + 12y - 7y - 84 = 0

y (y + 12) - 7(y + 12)

(y + 12) (y - 7) = 0

Therefore y = -12 or +7

It is mentioned that the numbers are positive whole numbers. So y can only be +7.

 Substituting y = 7 in equation 1, we get: x = 12

So the two numbers are 7 and 12.