Question

$\huge \fbox \red{❥ Question}$


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





(REQUIRED QUALITY ANSWER)​

Answer 1

Let ‘X’ and ‘Y’ be the two numbers.

Then By Question, We have,

X² - Y² = 180 ——————(1)

Y² = 8X ———————— (2)

Now, substituting the value of Y² from equation (2) in equation (1), we get,

X² - Y² = 180

X² - 8X = 180

X² - 18X + 10X = 180

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

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

Now, Either,

X-18 = 0 or X+10 = 0

X = 18 or X = -10 (ignored because negative)

Therefore, X = 18

Again, we know that,

Y² = 8X

Y = √8X

Y = √8*18

Y = √144

Y = 12

Hence, the two numbers are 12 and 18.

Answer 2

Required Answer:-

Given:

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

To find:

• The numbers.

Solution:

Let us assume that the numbers are x and y where y is the smaller number.

Difference between their squares is 180

➡ x² - y² = 180 — (i)

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

➡ y² = 8x [y is the smaller number and thus x is the larger number]

Substituting the value of y in the equation (i), we get,

➡ x² - 8x = 180

➡ x² - 18x + 10x - 180 = 0

➡ x(x - 18) + 10(x - 18) = 0

➡ (x + 10)(x - 18) = 0

By zero-product rule, we get,

➡ x = -10, 18

But x cannot be negative. (If it's negative, y becomes an imaginary number)

➡ x = 18

So,

➡ y² = 8x

➡ y = √(8 × 18)

➡ y = √(16 × 9)

➡ y = 3 × 4

➡ y = 12

Hence,

➡ x = 18

➡ y = 12

★ Hence the numbers are 18 and 12.

Answer:

• The numbers are 18 and 12.