Question

is y=-10x^2-140x-480 equivalent to y=-10(x+7)^2+10

Answer 1

To determine if the equations \( y = -10x^2 - 140x - 480 \) and \( y = -10(x + 7)^2 + 10 \) are equivalent, we can manipulate one of the equations to see if it can be transformed into the other.

Let's start with the second equation and expand it:

\[
y = -10(x + 7)^2 + 10
\]

First, we'll expand \( (x + 7)^2 \):

\[
(x + 7)^2 = x^2 + 14x + 49
\]

Now substitute this back into the equation:

\[
y = -10(x^2 + 14x + 49) + 10
\]

Next, distribute the \(-10\):

\[
y = -10x^2 - 140x - 490 + 10
\]

Now combine the constant terms:

\[
y = -10x^2 - 140x - 480
\]

Thus, we find:

\[
y = -10x^2 - 140x - 480
\]

This matches the first equation. Therefore, the two equations are indeed equivalent:

\[
y = -10x^2 - 140x - 480 \quad \text{is equivalent to} \quad y = -10(x + 7)^2 + 10.
\]