Two non-negative numbers are chosen such that their sum is 30. Find the numbers if the sum of their squares is to be a maximum.

Explanation to solve this would help !

m+n=30

m^2+n^2 = m^2+(30-m)^2 = 2m^2 - 60m + 900

Now, that's a parabola with vertex at (15,450)

Since the parabola opens upward, 450 is the minimum, and any maximum will be at the ends of the interval. We can't have m=0, so pick the numbers 1 and 29.

1^2+29^2 = 842

To find the two non-negative numbers such that their sum is 30, we can use algebraic expressions.

Let's assume the two numbers are x and y. According to the problem, their sum is 30. So, we can write the equation: x + y = 30.

To find the maximum value of the sum of their squares, we need to express the sum of squares in terms of one variable. Let's square both sides of the equation: (x + y)^2 = 30^2.

Expanding the left side of the equation: x^2 + 2xy + y^2 = 900.

Since our goal is to maximize the sum of squares, we need to simplify the equation further.

Expressing x^2 and y^2 in terms of their sum 's', we get:
(x + y)^2 = x^2 + 2xy + y^2 = (x^2 + y^2) + 2xy = s^2 + 2xy.

Now, we have the equation: s^2 + 2xy = 900.

Our task is to find the maximum value of s^2 (the sum of squares) by manipulating the equation.

To simplify this further, we need to eliminate the 'xy' term. We can do this by using the given information that the sum of the two numbers is 30.

We can rewrite the equation x + y = 30 as y = 30 - x.

Substituting y in terms of x, we get: s^2 + 2x(30 - x) = 900.

Expanding and rearranging the equation, we have: s^2 - 60x + 2x^2 = 900.

Now, we have a quadratic equation in terms of 's'.

To find the maximum value of s^2, we need to find the vertex of the quadratic equation.

The x-coordinate of the vertex can be found using the formula: x = -b/2a, where a = 2 and b = -60 in our equation.

Plugging in the values, we find x = -(-60) / (2*2) = 60 / 4 = 15.

Now that we have the x-coordinate of the vertex, we can substitute this value back into the equation y = 30 - x to find y.

Substituting x = 15 into y = 30 - x, we get y = 30 - 15 = 15.

Therefore, the two numbers are x = 15 and y = 15, and their sum of squares is 450 (15^2 + 15^2).