Question

The sum of two natural numbers is 12 and the sum of their squares is 74. Find the numbers.

Answer 1

Let the two natural numbers be $$x$$ and $$(12 - x)$$. 

According to the question: $$x^2 + (12 - x)^2 = 74$$

$$x^2 + 144 - 24x + x^2 = 74$$

$$2x^2 - 24x + 70 = 0$$ 

Dividing throughout by 2: $$x^2 - 12x + 35 = 0$$ 

Factoring: $$x^2 - 7x - 5x + 35 = 0$$

$$x(x - 7) - 5(x - 7) = 0$$ 

$$(x - 7)(x - 5) = 0$$

$$x = 7 \quad \text{or} \quad x = 5$$ 

If $$x = 7$$, the other number is $$12 - 7 = 5$$.

Hence, the two natural numbers are 5 and 7.