We need to find three real numbers $x$, $y$, and $z$ such that:

1. Their sum is $9$, i.e.,
   $x + y + z = 9$

2. The sum of their squares is as small as possible, i.e.,
   $x^2 + y^2 + z^2 \text{ is minimized}.$

Step 1: Use the constraint to express one variable in terms of the others

From the first condition $x + y + z = 9$, we can solve for $z$ in terms of $x$ and $y$:

$z = 9 - x - y$

Step 2: Substitute into the sum of squares

Now substitute $z = 9 - x - y$ into the sum of squares $x^2 + y^2 + z^2$:

$x^2 + y^2 + z^2 = x^2 + y^2 + (9 - x - y)^2$

Expanding the square:

= 81 - 18(x + y) + (x^2 + 2xy + y^2)$$ Thus, the expression becomes: $$x^2 + y^2 + z^2 = x^2 + y^2 + 81 - 18(x + y) + x^2 + 2xy + y^2$$ Simplify: $$= 2x^2 + 2y^2 + 2xy - 18(x + y) + 81$$ ### Step 3: Minimize the expression Now, we want to minimize the expression $$2x^2 + 2y^2 + 2xy - 18(x + y) + 81$$. To make the problem easier, observe that this expression is quadratic in $$x$$ and $$y$$, so the minimum value occurs when the variables $$x$$ and $$y$$ are balanced. ### Step 4: Symmetry Given that the expression is symmetric in $$x$$ and $$y$$, the sum of squares is minimized when $$x = y = z$$. This is because when the values are balanced, the sum of squares is smaller. Thus, set $$x = y = z$$. From the constraint $$x + y + z = 9$$, we get: $$3x = 9 \quad \Rightarrow \quad x = 3$$ So, $$x = y = z = 3$$. ### Final Answer: The three numbers are $$x = 3$$, $$y = 3$$, and $$z = 3$$. This gives the minimal sum of squares. Would you like more details on this solution? Here are some related questions for further exploration: 1. How can we prove that $$x = y = z$$ gives the minimal sum of squares? 2. What if the constraint was different, such as $$x + y + z = 12$$? Would the result still be $$x = y = z$$? 3. How would you solve this problem using Lagrange multipliers? 4. What happens if the condition is changed to $$x + y + z = 0$$? 5. What is the geometric interpretation of minimizing the sum of squares in this context? **Tip**: When trying to minimize a sum of squares under a linear constraint, symmetry is often a helpful indicator of the optimal solution.