Let $a,$ $b,$ $c$ be real numbers such that $a + b + c = 0.$  Find the set of all possible values of $ab + ac + bc.$
Squaring the equation $a + b + c = 0,$ we get
\[a^2 + b^2 + c^2 + 2(ab + ac + bc) = 0.\]Hence, $2(ab + ac + bc) = -(a^2 + b^2 + c^2) \le 0,$ so
\[ab + ac + bc \le 0.\]Equality occurs when $a = b = c = 0.$

Now, set $c = 0,$ so $a + b = 0,$ or $b = -a.$  Then
\[ab + ac + bc = ab = -a^2\]can take on all nonpositive values.  Therefore, the set of all possible values of $ab + ac + bc$ is $\boxed{(-\infty,0]}.$
(-\iny,0]
Level 5
Intermediate Algebra
Problem: Let $a,$ $b,$ $c$ be real numbers such that $a + b + c = 0.$  Find the set of all possible values of $ab + ac + bc.$
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the set of all possible values of \( ab + ac + bc \), we start by expressing \( c \) in terms of \( a \) and \( b \) using the condition \( a + b + c = 0 \). This gives us \( c = -a - b \).

Next, we substitute \( c \) into the expression \( ab + ac + bc \):

\[
ab + ac + bc = ab + a(-a - b) + b(-a - b)
\]

Expanding this, we have:

\[
ab - a^2 - ab - ab - b^2 = -a^2 - ab - b^2
\]

This simplifies to:

\[
-ab - a^2 - b^2
\]

Now, we can rewrite this expression as:

\[
-(a^2 + ab + b^2)
\]

To analyze the expression \( a^2 + ab + b^2 \), we recognize that it is a quadratic in terms of \( a \) and \( b \). The discriminant of this quadratic must be non-negative for \( a \) and \( b \) to yield real values. The discriminant \( D \) of \( a^2 + ab + b^2 \) can be calculated as follows:

\[
D = b^2 - 4(1)(b^2) = b^2 - 4b^2 = -3b^2
\]

Since \( D \) is non-negative only when \( b = 0 \), we conclude that \( a^2 + ab + b^2 \) is minimized when \( b = 0 \). In this case, we have:

\[
a^2 + 0 + 0 = a^2
\]

Thus, the minimum value of \( a^2 + ab + b^2 \) is \( 0 \) when \( a = 0 \) and \( b = 0 \). 

As \( a^2 + ab + b^2 \) can take any non-negative value, the expression \( -(a^2 + ab + b^2) \) can take any non-positive value. Therefore, the set of all possible values of \( ab + ac + bc \) is:

\[
(-\infty, 0]
\]<|END_OF_LATENT|> The final answer is $(-\iny,0]$.