Question

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

Answer 1

Given:

• Difference of their squares. i.e. 180 ..( 1 )
• Square of smaller number is 8 times the larger number ... ( 2)

To Find:

• The numbers.

Solution:

• let smaller number = a
• let the larger number = b

( using ( 2 ) )

⟹ a² = 8b

⟹ a = +√8b

( using ( 1 ) )

⟹ b² - a² = 180

( substitute +√8b in place of a )

⟹ b² - ( +√8b )² = 180

⟹ b² - 8b = 180

⟹ b² - 8b - 180 = 0

( by splitting the middle term)

⟹ b² + 10b - 18b - 180 = 0

⟹ b( b + 10 ) - 18 ( b + 10 ) = 0

⟹ ( b + 10 )(b - 18) = 0

b + 10 = 0

b = - 10

or

b - 18 = 0

b = 18

zeroes of the equation are (-10) and 18.

if (-10) is the larger number (b) :

a = +√8b

a = + √8×(-10)

a = + √ -80

it is not possible because we can't find the square roots of negative Numbers.

if 18 is the larger number (b) :

a = +√8b

a = +√8(18)

a = +√144

a = + 12

So, the numbers are

18 and 12 or 18 and -12

Answer 2

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