Question

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find two numbers.

A) 10 & 12
B) 10 & 18
C) 12 & -18
D) -12 & 18

Answer 1

Answer : 18 and 12

Explanation :

Let the numbers be x and y

Larger number be x and smaller be y

The difference between the squares of the numbers = 180

According to the problem,

x² - y² = 180 --------(1)

The square of the smaller number is 8 times the larger number.

y² = 8x --------(2)

Substitute eq - (2) and eq - (1)

Then we get the quadratic equation,

x² - 8x - 180 = 0

Split the middle terms.

x² - 18x + 10x - 180 = 0

x(x - 18) + 10(x - 18) = 0

x - 18 = 0 ; x + 10 = 0

x = 18 ; x = -10

We should take the positive value.

Thus, the larger number is 18.

Then the smaller number :

Substitute 18 in eq - (2)

y² = 8(18)

y² = 144

y = √144

y = 12

Therefore, the numbers are 18 and 12.

Answer 2

$\mathfrak{\large{\underline{\underline{Answer:-}}}}$

$\boxed{\bf{12\:and\:18}}$

$\mathfrak{\large{\underline{\underline{Explanation:-}}}}$

Given :-

Difference of squares of of two numbers = 180

Square of smaller number is 8 times the larger number

To find :- Two numbers

Solution :-

Let the numbers be x and y  

Squares of two numbers = x² and y²

Given that Square of smaller number is 8 times the larger number

⇒ y² = 8(x)

⇒ y² = 8x

Given that Difference of squares of of two numbers = 180

Equation formed :- x² - y² = 180

⇒ x² - 8x = 180

[Since y² = 8x]

Split the middle term:

⇒ x² - 8x - 180 = 0

180 = 2 × 2 × 3 × 3 × 5

x² + 10x - 18x - 180 = 0

⇒ x(x + 10) - 18(x + 10) = 0

⇒ (x +  10)(x - 18) = 0

Now equate the products to 0

⇒ x + 10 = 0 or x - 18 = 0

⇒ x = - 10 or x = 18

⇒ x = - 10 or + 18

Then y² = 8(-10)

y = √(- 80) not possible

Or y² = 8(18) = 144

y = 12

$\mathfrak{\large{\underline{\underline{Verification:-}}}}$

⇒ 18² - 12² = 180

⇒ 324 - 144 = 180

⇒ 180 = 180

Hence value of x is 18 and y is 12

$\boxed{\bf{12\:and\:18}}$