Question

the sum of the squares of two integers is 306. if the square of the larger number is 25 times the smaller integer . find the integers

Answer 1

Larger integer = 15 and smaller number = 9

Given :

• The sum of the squares of two integers is 306

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

To find :

The integers

Solution :

Step 1 of 2 :

Form the equation to find the integers

Let smaller integer = x and larger integer = y

Then , y² = 25x - - - - - (1)

Also the sum of the squares of two integers is 306

x² + y² = 306

⇒ x² + 25x = 306

Step 2 of 2 :

Find the integers

$\displaystyle \sf {x}^{2} + 25x = 306$

$\displaystyle \sf{ \implies }{x}^{2} + 25x - 306 = 0$

$\displaystyle \sf{ \implies }{x}^{2} + (34 - 9)x - 306 = 0$

$\displaystyle \sf{ \implies }{x}^{2} + 34x - 9x - 306 = 0$

$\displaystyle \sf{ \implies }x(x + 34) - 9(x + 34) = 0$

$\displaystyle \sf{ \implies }(x + 34) (x - 9) = 0$

x + 34 = 0 gives x = - 34

Consequently y² = - 25 × 34

Which is absurd

Again x - 9 = 0 gives x = 9

Consequently ,

y² = 25 × 9

⇒ y² = 225

⇒ y = 15

Hence Larger integer = 15 and smaller number = 9

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