Let $A$ be the vertex of the graph of the equation $y=x^2 - 2x + 3 $. Let $B$ be the vertex of the graph of the equation $y=x^2 + 4x + 10 $. What is the distance between $A$ and $B$?
Completing the square in each equation results in the equations $y=(x - 1)^2 + 2 $ and $y=(x + 2)^2 + 6$. Thus, $A = (1, 2)$ and $B = (-2, 6)$. We can then find the distance between $A$ and $B$ to be $\sqrt{(1-(-2))^2 + (2-6)^2} = \sqrt{9+16} =\boxed{5}$.
5
Level 4
Algebra
Problem: Let $A$ be the vertex of the graph of the equation $y=x^2 - 2x + 3 $. Let $B$ be the vertex of the graph of the equation $y=x^2 + 4x + 10 $. What is the distance between $A$ and $B$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the vertices of the given quadratic equations, we use the vertex formula for a parabola represented by the equation $y = ax^2 + bx + c$. The x-coordinate of the vertex is given by $x = -\frac{b}{2a}$. 

For the first equation $y = x^2 - 2x + 3$, we identify $a = 1$, $b = -2$, and $c = 3$. Applying the vertex formula, we calculate the x-coordinate of vertex $A$:

x-coordinate of $A = -\frac{-2}{2 \cdot 1} = \frac{2}{2} = 1.

Next, we substitute $x = 1$ back into the equation to find the y-coordinate of vertex $A$:

y-coordinate of $A = (1)^2 - 2(1) + 3 = 1 - 2 + 3 = 2.

Thus, the coordinates of vertex $A$ are $(1, 2)$.

For the second equation $y = x^2 + 4x + 10$, we identify $a = 1$, $b = 4$, and $c = 10$. Using the vertex formula again, we calculate the x-coordinate of vertex $B$:

x-coordinate of $B = -\frac{4}{2 \cdot 1} = -\frac{4}{2} = -2.

Now, we substitute $x = -2$ back into the equation to find the y-coordinate of vertex $B$:

y-coordinate of $B = (-2)^2 + 4(-2) + 10 = 4 - 8 + 10 = 6.

Thus, the coordinates of vertex $B$ are $(-2, 6)$.

To find the distance between points $A(1, 2)$ and $B(-2, 6)$, we use the distance formula:

Distance = $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Substituting the coordinates of $A$ and $B$:

Distance = $\sqrt{((-2) - 1)^2 + (6 - 2)^2} = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

The final answer is $5$.<|END_OF_LATENT|> The final answer is $5$.