Question

Fill in the parts-whole model for the following equation.
left parenthesis, minus, 3, x, squared, minus, 10, x, plus, 1, right parenthesis, plus, left parenthesis, start underline, end underline, right parenthesis, equals, minus, 10, x, squared, minus, 18, x, plus, 10
(−3x
2
−10x+1)+(
​
)=−10x
2
−18x+10

Answer 1

To fill in the parts-whole model for the equation:

\[
(-3x^2 - 10x + 1) + ( \underline{ } ) = -10x^2 - 18x + 10
\]

we need to find the expression that, when added to \(-3x^2 - 10x + 1\), results in \(-10x^2 - 18x + 10\).

To do this, we can set up an equation:

Let \(P = -3x^2 - 10x + 1\) and let \(Q\) be the expression we need to find. Then we have:

\[
P + Q = -10x^2 - 18x + 10
\]

We can find \(Q\) by rearranging the equation:

\[
Q = (-10x^2 - 18x + 10) - P
\]
\[
Q = (-10x^2 - 18x + 10) - (-3x^2 - 10x + 1)
\]
\[
Q = -10x^2 - 18x + 10 + 3x^2 + 10x - 1
\]
\[
Q = (-10x^2 + 3x^2) + (-18x + 10x) + (10 - 1)
\]
\[
Q = -7x^2 - 8x + 9
\]

So, the expression we need to fill in is:

\[
\underline{-7x^2 - 8x + 9}
\]

Thus, the complete equation will be:

\[
(-3x^2 - 10x + 1) + (-7x^2 - 8x + 9) = -10x^2 - 18x + 10
\]