Which sum or difference is modelled by the algebra tiles? The figure shows the elimination of four red tiles and four blue tiles in which two tiles numbered with 1, two tiles with a minus square of x, and two tiles with x.
A. (x^(2) + 4x – 2) + (x^(2) – 2x – 4) = 2x^(2) + 2x + 2
B. (x^(2) + 4x – 2) + (x^(2) + 2x + 4) = 2x^(2) + 2x + 2
C. (x^(2) + 4x – 2) – (-x^(2) + 2x – 4) = 2x^(2) + 2x +
The Correct Answer and Explanation is:
The correct answer is:
A. (x² + 4x – 2) + (x² – 2x – 4) = 2x² + 2x – 6
However, this doesn’t match the result given in A, which states the answer is 2x² + 2x + 2. This indicates a mismatch between the expressions and their calculated result.
Let’s break this down carefully.
Interpreting Algebra Tiles
Algebra tiles are a visual method to model expressions. Typically:
- Large squares represent x² terms.
- Rectangles represent x or -x terms.
- Small squares represent constants: +1 or -1.
In the described figure, it says:
- Four red tiles and four blue tiles are eliminated. This implies using zero pairs, which are combinations of a positive and a negative tile of the same type (like +x and -x or +1 and -1).
- Tiles shown:
- Two x² tiles (probably +x²).
- Two -x tiles.
- Two x tiles.
- Some constants (likely two +1s and two -1s).
So, the figure shows:
- 2 x² tiles.
- 2 +x tiles and 2 -x tiles cancel (net 0x).
- 2 +1 tiles and 2 -1 tiles cancel (net 0).
Hence, the remaining expression is:
2x² + 0x + 0 = 2x²
This contradicts the answer provided in all the options.
Let’s analyze each option:
Option A:
(x² + 4x – 2) + (x² – 2x – 4)
Add like terms:
- x² + x² = 2x²
- 4x – 2x = 2x
- -2 – 4 = -6
So, the sum is: 2x² + 2x – 6
This is NOT 2x² + 2x + 2, so Option A is incorrect.
Option B:
(x² + 4x – 2) + (x² + 2x + 4)
Add like terms:
- x² + x² = 2x²
- 4x + 2x = 6x
- -2 + 4 = 2
Sum: 2x² + 6x + 2
Not 2x² + 2x + 2, so Option B is incorrect.
Option C:
(x² + 4x – 2) – (-x² + 2x – 4)
Distribute the negative sign:
- x² + 4x – 2 + x² – 2x + 4
Combine like terms:
- x² + x² = 2x²
- 4x – 2x = 2x
- -2 + 4 = 2
Final result: 2x² + 2x + 2
This matches the result shown.
✅ Correct Answer: C. (x² + 4x – 2) – (-x² + 2x – 4) = 2x² + 2x + 2
Explanation (300+ words):
Algebra tiles are a powerful visual aid in understanding the addition and subtraction of polynomials. Each type of tile represents a term in the expression: large square tiles for $x^2$, rectangular tiles for $x$, and small square tiles for constants like +1 or -1. Positive values are usually colored blue, while negative values are red.
In this case, the diagram shows the elimination (cancellation) of equal numbers of red and blue tiles, meaning pairs like $+x$ and $-x$, or $+1$ and $-1$, are being removed. After eliminating 4 red and 4 blue tiles (which likely include 2 +x and 2 -x, and 2 +1 and 2 -1), the only visible tiles remaining are 2 $x^2$ tiles, meaning we are left with $2x^2$.
Now, let’s analyze Option C:
(x² + 4x – 2) – (-x² + 2x – 4)
First, apply the distributive property to remove the parentheses in the subtraction:
- $(x^2 + 4x – 2) – (-x^2 + 2x – 4)$
- becomes $x^2 + 4x – 2 + x^2 – 2x + 4$
Next, combine like terms:
- $x^2 + x^2 = 2x^2$
- $4x – 2x = 2x$
- $-2 + 4 = 2$
This results in:
$2x^2 + 2x + 2$
This matches the expression modeled by the tiles. Thus, Option C correctly represents the sum shown in the algebra tile