The problem states:

"The sum of the squares of two consecutive natural numbers is 313. Find the numbers."

Solution:

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

1. The square of $x$ is $x^2$, and the square of $x+1$ is $(x+1)^2$.
2. The sum of their squares is given as: $x^2 + (x+1)^2 = 313$
3. Expanding $(x+1)^2$: $x^2 + x^2 + 2x + 1 = 313$
4. Combine like terms: $2x^2 + 2x + 1 = 313$
5. Subtract 313 from both sides: $2x^2 + 2x - 312 = 0$
6. Simplify by dividing through by 2: $x^2 + x - 156 = 0$

Solve the Quadratic Equation:

$x^2 + x - 156 = 0$ Using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = 1$, $c = -156$.

1. Calculate the discriminant: $\Delta = b^2 - 4ac = 1^2 - 4(1)(-156) = 1 + 624 = 625$
2. Solve for $x$: $x = \frac{-1 \pm \sqrt{625}}{2}$ $x = \frac{-1 \pm 25}{2}$
3. Two solutions: $x = \frac{-1 + 25}{2} = \frac{24}{2} = 12$ $x = \frac{-1 - 25}{2} = \frac{-26}{2} = -13$

Since $x$ must be a natural number, $x = 12$.

Final Answer:

The two consecutive natural numbers are 12 and 13.

Verification:

$12^2 + 13^2 = 144 + 169 = 313$

The solution is correct.

Would you like a step-by-step explanation or help with a related concept?

Related Questions:

1. What are consecutive numbers, and how do they differ from consecutive even/odd numbers?
2. How does the quadratic formula work, and when is it used?
3. Can this problem be solved without using the quadratic formula? If so, how?
4. What happens if the given sum is not a perfect square?
5. Why is it necessary to reject negative values in this context?

Tip:

When solving quadratic equations, always verify the solution by substituting the values back into the original equation!