Question

# The difference of the squares of two natural numbers is 45. The square of the smaller number is four times the larger number. Find the numbers.

Solution

## Let the smaller natural number be x and larger natural number be y Hence x2=4y → (1) Given y2–x2=45 ⇒ y2–4y=45 ⇒y2–4y–45=0 ⇒ y2–9y+5y–45=0 ⇒ y(y–9)+5(y–9)=0 ⇒(y–9)(y+5)=0 ⇒(y–9)=0 or (y+5)=0 ∴y=9ory=−5 But y is natural number, hence y ≠ - 5 Therefore, y = 9 Equation (1) becomes, x2=4(9)=36 ∴ x = 6 Thus the two natural numbers are 6 and 9. MathematicsSecondary School Mathematics XStandard X

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