Question

Find two consecutive natural numbers, the sum of whose squares is 145.

Answer 1

Let the two consecutive natural numbers be $$x$$ and $$(x + 1)$$. 

According to the question: $$x^2 + (x + 1)^2 = 145$$

$$x^2 + x^2 + 2x + 1 = 145$$

$$2x^2 + 2x - 144 = 0$$

Dividing the entire equation by 2: $$x^2 + x - 72 = 0$$

Factoring: $$x^2 + 9x - 8x - 72 = 0$$

$$x(x + 9) - 8(x + 9) = 0$$ 

$$(x + 9)(x - 8) = 0$$

$$x = -9 \quad \text{or} \quad x = 8$$ 

Since $$x$$ is a natural number, reject $$x = -9$$.

Thus, $$x = 8$$ and the next consecutive number is $$8 + 1 = 9$$.

Hence, the required natural numbers are 8 and 9.