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.
​

Answer 1

Answer:

We have,

Let the large number is =x

Square of smaller number is =8x

Now, According to given question,

x

2

−8x=180

⇒x

2

−8x−180=0

⇒x

2

−(18−10)x−180=0

⇒x

2

−18x+10x−180=0

⇒x(x−18)+10(x−18)=0

⇒(x−18)(x+10)=0

⇒x−18=0,x+10=0

⇒x=18,x=−10

⇒x=−10,18

Either

x=−10 and x=18

x=18 is true (Positive value)

Now, square of small number

=18x=18×8=144

144

=12

Hence, this is the answer.

Answer 2

Answer:

Numbers are 18 and 12

Let the larger square number be x² so the smaller square number be x² - 180

Given,

The square of the smaller number is 8

times the larger number.

Hence,

(Smaller number)² = 8 × larger number

→ x² - 180 = 8 × x

→ x² - 180 = 8x

→ x² - 8x - 180 = 0

→ x² + 10x - 18x - 180 = 0

→ x(x + 10) - 18(x + 10) = 0

→ (x - 18)(x + 10) = 0

-----------------------------

x - 18 = 0

→ x = 18

or

x + 10 = 0

→ x = -10

But x = -10 doesn't satisfy our equation, hence it needs to be rejected.

[ Note : If we accept negative value, the larger number will become the smaller number and vice versa and hence it will not satisfy our problem, therefore it needs to be rejected]

Larger number = x = 18

(Smaller number)² = 8 × x = 8 × 18 = 144

(Smaller number)² = 144

Smaller number = 12 [on taking the square root]