To find the product of \((x - 5)^2\) using algebra tiles, we can interpret the expression as \((x - 5)(x - 5)\). We'll use the following steps to visualize it:
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Understanding the expression: The expression \((x - 5)\) represents a binomial, meaning it consists of two parts: \(x\) and \(-5\).
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Setting up the algebra tiles:
- Consider \(x\) as a large tile representing \(x\).
- The \(-5\) can be represented as five smaller tiles, each representing \(-1\).
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Arrange the tiles:
- On a rectangular grid, arrange two \(x\) tiles horizontally on the top row.
- Below the \(x\) tiles, place two groups of \(-5\) tiles extending out from each \(x\) tile.
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Finding the area: The area of the arrangement represents the product \((x - 5)(x - 5)\).
- First, multiply the \(x\) terms:
- \(x \cdot x = x^2\)
- Next, calculate the interaction:
- \(x \cdot -5\) gives \(-5x\)
- The second \(x \cdot -5\) gives another \(-5x\)
- Finally, the product of the \(-5\) tiles:
- \(-5 \cdot -5 = 25\)
- First, multiply the \(x\) terms:
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Combine the results:
- From the above, you get:
- \(x^2\) from the \(x \cdot x\)
- \(-5x - 5x = -10x\) from the two interactions
- And finally, \(+25\) from the \(-5 \cdot -5\)
- From the above, you get:
Putting it all together, you get the expression:
\[ (x - 5)^2 = x^2 - 10x + 25 \]
Thus, the product of \((x - 5)^2\) is:
\[ x^2 - 10x + 25 \]
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