Question

The sum of the squares of two positive integers is 208. If the square of the

larger number is 18 times the smaller number, find the numbers​

Answer 1

✪ x = 12 , y = 8 ✪

☯ Let the two numbers be x,y , and here let the great no. be x . So , by the given information ,

♦ x² + y² = 208 ...eq(1)

♦ x² = 18y

So , now substituting x² = 18y in eq(1)

⇒ 18y + y² = 208

⇒ y² + 18y - 208 = 0

⇒ y² + 26y - 8y + 208 = 0

⇒ y ( y + 26 ) - 8 ( y + 26 ) = 0

⇒ ( y + 26 ) ( y - 8 ) = 0

⇒ y = - 26 or 8

Here , given that no.'s are positive , so here y = 8 . Now , finding x

→ x² = 18y

→ x = √(18y)

→ x = √(9×2)(y)

→ x = 3√(2y)

Now , substituting y = 8

=> x = 3√(2×8)

=> x = 3√16

=> x = 3√4²

=> x = 3 × 4

=> x = 12

Hence , x = 12 & y = 8 .

Answer 2

$\Large{\bf{\pink{\underline{Let,}}}}$

• P & Q are two positive integers.

• P is greater than Q.

$\Large{\bf{\green{\underline{GiVeN,}}}}$

• The sum of the squares of two positive integers is 208.

$\longmapsto\:\:\bf\blue{P^2\:+\:Q^2\:=\:208}--(1)$

$\bf\red{And,}$

• The square of the larger number is 18 times to the smaller number.

$\longmapsto\:\:\bf\orange{P^2\:=\:18Q\:}--(2)$

$\Large{\bf{\purple{\underline{To\:FiNd,}}}}$

• The numbers.

$\Large{\bf{\pink{\underline{CaLcUlAtIoN,}}}}$

↝ Putting the value of P² in equation (1), we get

$:\implies\:\:\bf{18Q\:+\:Q^2\:=\:208}$

$:\implies\:\:\bf{Q^2\:+\:18Q\:-\:208\:=\:0}$

$:\implies\:\:\bf{Q^2\:+\:26Q\:-\:8Q\:-\:208\:=\:0}$

$:\implies\:\:\bf{Q\:(Q\:+\:26)\:-8\:(Q\:-\:26)\:=\:0}$

$:\implies\:\:\bf{(Q\:+\:26)\:(Q\:-\:8)\:=\:0}$

$:\implies\:\:\bf{Q\:+\:26\:=\:0\:~~~~or~~~~\:Q\:-\:8\:=\:0}$

$:\implies\:\:\bf{Q\:=\:-\:26\:~~~~or~~~~\:Q\:=\:8}$

[NOTE ➛ It is given that both numbers are positive integers.]

$:\implies\:\:\bf\orange{Q\:=\:8}$

$\bf\red{Now,}$

↝ Putting the value of Q in the equation (2), we get

$:\implies\:\:\bf{P^2\:=\:18\times{8}\:}$

$:\implies\:\:\bf{P^2\:=\:144\:}$

$:\implies\:\:\bf{P\:=\:\sqrt{144}\:}$

$:\implies\:\:\bf{P\:=\:\pm\:12\:}$

[NOTE ➛ It is given that number is a positive integer.]

$:\implies\:\:\bf\green{P\:=\:12\:}$

$\Large\bf\blue{Therefore,}$

The two positive integers are 12 & 8.