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

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#### Solution

Let the larger and smaller number be x and y respectively.

According to the question,

x^{2} - y^{2} = 180 and y^{2} = 8x

⇒ x^{2} - 8x = 180

⇒ x^{2} - 8x - 180 = 0

⇒ x^{2}- 18x + 10x - 180 = 0

⇒ x(x - 18) +10(x - 18) = 0

⇒ (x - 18)(x + 10) = 0

⇒ x = 18, -10

However, the larger number cannot be negative as 8 times of the larger number will be negative and hence, the square of the smaller number will be negative which is not possible.

Therefore, the larger number will be 18 only.

x = 18

∴ y^{2} = 8x = 8 × 18 = 144

⇒ y = `±sqrt144 = ±12`

∴ Smaller number = ±12

Therefore, the numbers are 18 and 12 or 18 and - 12.

Concept: Solutions of Quadratic Equations by Completing the Square

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