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,

x2−8x=180

⇒x2−8x−180=0

⇒x2−(18−10)x−180=0

⇒x2−18x+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

Let the larger number is taken as = x

Then the square of the smaller number = 8 times the larger number = 8x and the square of the larger number = x

As per the question,

x – 8x = 180

⇒ x – 8x – 180 = 0

⇒ x – 18x + 10x – 180 = 0

⇒ x(x – 18) + 10(x – 18) = 0

⇒ (x – 18) (x + 10) = 0

⇒ x – 18 = 0 or x + 10 = 0

⇒ x = 18 or x = -10

Thus, the larger number = 18 or -10

Then, the square of the smaller number = 8(18) or 8(-10) = 144 or -80

The square of a number can’t be negative, so, the square of a smaller number = 144

Hence, the smaller number = sqrt(144) = 12

The numbers are 12 and 18