Question

Fill in the parts-whole model for the following equation.
left parenthesis, start underline, end underline, right parenthesis, minus, left parenthesis, minus, 7, x, squared, minus, 2, x, plus, 4, right parenthesis, equals, 5, x, squared, plus, 10, x, minus, 4
(
​
)−(−7x
2
−2x+4)=5x
2
+10x−4

Answer 1

To fill in the parts-whole model for the equation

\[
( \underline{\quad} ) - (-7x^2 - 2x + 4) = 5x^2 + 10x - 4
\]

let's first rewrite the equation to clarify the left side:

\[
( \underline{\quad} ) + (7x^2 + 2x - 4) = 5x^2 + 10x - 4
\]

Here, we can see that we have a "whole" on the left side, which consists of the expression on the left side of the equation when we account for the subtraction of the negative expression.

We can organize the parts-whole model as follows:

- **Whole:** \( \underline{\quad} \) = ? (This is what we're solving for)
- **Part 1 (added value):** \( (7x^2 + 2x - 4) \)
- **Part 2 (resulting sum):** \( (5x^2 + 10x - 4) \)

To find the whole, we can set up the equation from above:

\[
\text{Whole} = \text{Part 2} - \text{Part 1}
\]

Substituting in the expressions:

\[
\text{Whole} = (5x^2 + 10x - 4) - (7x^2 + 2x - 4)
\]

Now we simplify:

1. Distribute negative:
\[
\text{Whole} = 5x^2 + 10x - 4 - 7x^2 - 2x + 4
\]

2. Combine like terms:
- The \(x^2\) terms: \(5x^2 - 7x^2 = -2x^2\)
- The \(x\) terms: \(10x - 2x = 8x\)
- The constant terms: \(-4 + 4 = 0\)

Putting it all together, we find:

\[
\text{Whole} = -2x^2 + 8x
\]

So filling in the parts-whole model:

- **Whole:** \( -2x^2 + 8x \)
- **Part 1:** \( 7x^2 + 2x - 4 \)
- **Part 2:** \( 5x^2 + 10x - 4 \)

This gives us the complete parts-whole model for the equation.