Solved please help with this I Heart Factoring worksheet. | Chegg.com - Free Printable

The task involves factoring quadratic expressions and matching them with their corresponding factored forms. Below, I will solve each problem step by step and provide the correct matches.

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Step-by-Step Solutions:

#### Problem 1: \( x^2 - 2x + 1 \)
This is a perfect square trinomial:
\[
x^2 - 2x + 1 = (x - 1)^2
\]
Match: \( (x - 1)^2 \) (Piece labeled "S")

#### Problem 2: \( x^2 - 6x - 16 \)
Factor by finding two numbers that multiply to \(-16\) and add to \(-6\). These numbers are \(-8\) and \(+2\):
\[
x^2 - 6x - 16 = (x - 8)(x + 2)
\]
Match: \( (x - 8)(x + 2) \) (Piece labeled "T")

#### Problem 3: \( x^2 + 10x + 16 \)
Factor by finding two numbers that multiply to \(16\) and add to \(10\). These numbers are \(8\) and \(2\):
\[
x^2 + 10x + 16 = (x + 8)(x + 2)
\]
Match: \( (x + 8)(x + 2) \) (Piece labeled "U")

#### Problem 4: \( x^2 - 13x + 42 \)
Factor by finding two numbers that multiply to \(42\) and add to \(-13\). These numbers are \(-7\) and \(-6\):
\[
x^2 - 13x + 42 = (x - 7)(x - 6)
\]
Match: \( (x - 7)(x - 6) \) (Piece labeled "V")

#### Problem 5: \( x^2 + 8x - 9 \)
Factor by finding two numbers that multiply to \(-9\) and add to \(8\). These numbers are \(9\) and \(-1\):
\[
x^2 + 8x - 9 = (x + 9)(x - 1)
\]
Match: \( (x + 9)(x - 1) \) (Piece labeled "W")

#### Problem 6: \( x^2 - 8x + 2 \)
This quadratic does not factor nicely over the integers. It remains as is:
\[
x^2 - 8x + 2
\]
Match: \( x^2 - 8x + 2 \) (Piece labeled "X")

#### Problem 7: \( x^2 - x - 6 \)
Factor by finding two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(+2\):
\[
x^2 - x - 6 = (x - 3)(x + 2)
\]
Match: \( (x - 3)(x + 2) \) (Piece labeled "Y")

#### Problem 8: \( x^2 + 15x + 50 \)
Factor by finding two numbers that multiply to \(50\) and add to \(15\). These numbers are \(10\) and \(5\):
\[
x^2 + 15x + 50 = (x + 10)(x + 5)
\]
Match: \( (x + 10)(x + 5) \) (Piece labeled "Z")

#### Problem 9: \( x^2 - 3x - 2 \)
This quadratic does not factor nicely over the integers. It remains as is:
\[
x^2 - 3x - 2
\]
Match: \( x^2 - 3x - 2 \) (Piece labeled "A")

#### Problem 10: \( x^2 - 5x - 24 \)
Factor by finding two numbers that multiply to \(-24\) and add to \(-5\). These numbers are \(-8\) and \(+3\):
\[
x^2 - 5x - 24 = (x - 8)(x + 3)
\]
Match: \( (x - 8)(x + 3) \) (Piece labeled "B")

#### Problem 11: \( x^2 - 12x + 36 \)
This is a perfect square trinomial:
\[
x^2 - 12x + 36 = (x - 6)^2
\]
Match: \( (x - 6)^2 \) (Piece labeled "C")

#### Problem 12: \( x^2 - 8x + 16 \)
This is a perfect square trinomial:
\[
x^2 - 8x + 16 = (x - 4)^2
\]
Match: \( (x - 4)^2 \) (Piece labeled "D")

#### Problem 13: \( x^2 + x - 2 \)
Factor by finding two numbers that multiply to \(-2\) and add to \(1\). These numbers are \(2\) and \(-1\):
\[
x^2 + x - 2 = (x + 2)(x - 1)
\]
Match: \( (x + 2)(x - 1) \) (Piece labeled "E")

#### Problem 14: \( x^2 - 3x + 2 \)
Factor by finding two numbers that multiply to \(2\) and add to \(-3\). These numbers are \(-2\) and \(-1\):
\[
x^2 - 3x + 2 = (x - 2)(x - 1)
\]
Match: \( (x - 2)(x - 1) \) (Piece labeled "F")

#### Problem 15: \( x^2 + 6x + 9 \)
This is a perfect square trinomial:
\[
x^2 + 6x + 9 = (x + 3)^2
\]
Match: \( (x + 3)^2 \) (Piece labeled "G")

#### Problem 16: \( x^2 - 9 \)
This is a difference of squares:
\[
x^2 - 9 = (x - 3)(x + 3)
\]
Match: \( (x - 3)(x + 3) \) (Piece labeled "H")

#### Problem 17: \( x^2 + 5x + 6 \)
Factor by finding two numbers that multiply to \(6\) and add to \(5\). These numbers are \(3\) and \(2\):
\[
x^2 + 5x + 6 = (x + 3)(x + 2)
\]
Match: \( (x + 3)(x + 2) \) (Piece labeled "I")

#### Problem 18: \( x^2 - 16 \)
This is a difference of squares:
\[
x^2 - 16 = (x - 4)(x + 4)
\]
Match: \( (x - 4)(x + 4) \) (Piece labeled "J")

#### Problem 19: \( x^2 + 6x + 8 \)
Factor by finding two numbers that multiply to \(8\) and add to \(6\). These numbers are \(4\) and \(2\):
\[
x^2 + 6x + 8 = (x + 4)(x + 2)
\]
Match: \( (x + 4)(x + 2) \) (Piece labeled "K")

#### Problem 20: \( x^2 + 5x + 9 \)
This quadratic does not factor nicely over the integers. It remains as is:
\[
x^2 + 5x + 9
\]
Match: \( x^2 + 5x + 9 \) (Piece labeled "L")

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Final Matches:

1. \( x^2 - 2x + 1 \) → \( (x - 1)^2 \) (S)
2. \( x^2 - 6x - 16 \) → \( (x - 8)(x + 2) \) (T)
3. \( x^2 + 10x + 16 \) → \( (x + 8)(x + 2) \) (U)
4. \( x^2 - 13x + 42 \) → \( (x - 7)(x - 6) \) (V)
5. \( x^2 + 8x - 9 \) → \( (x + 9)(x - 1) \) (W)
6. \( x^2 - 8x + 2 \) → \( x^2 - 8x + 2 \) (X)
7. \( x^2 - x - 6 \) → \( (x - 3)(x + 2) \) (Y)
8. \( x^2 + 15x + 50 \) → \( (x + 10)(x + 5) \) (Z)
9. \( x^2 - 3x - 2 \) → \( x^2 - 3x - 2 \) (A)
10. \( x^2 - 5x - 24 \) → \( (x - 8)(x + 3) \) (B)
11. \( x^2 - 12x + 36 \) → \( (x - 6)^2 \) (C)
12. \( x^2 - 8x + 16 \) → \( (x - 4)^2 \) (D)
13. \( x^2 + x - 2 \) → \( (x + 2)(x - 1) \) (E)
14. \( x^2 - 3x + 2 \) → \( (x - 2)(x - 1) \) (F)
15. \( x^2 + 6x + 9 \) → \( (x + 3)^2 \) (G)
16. \( x^2 - 9 \) → \( (x - 3)(x + 3) \) (H)
17. \( x^2 + 5x + 6 \) → \( (x + 3)(x + 2) \) (I)
18. \( x^2 - 16 \) → \( (x - 4)(x + 4) \) (J)
19. \( x^2 + 6x + 8 \) → \( (x + 4)(x + 2) \) (K)
20. \( x^2 + 5x + 9 \) → \( x^2 + 5x + 9 \) (L)

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Final Answer:

\[
\boxed{
\begin{aligned}
1 &\to S, \, 2 \to T, \, 3 \to U, \, 4 \to V, \, 5 \to W, \, 6 \to X, \, 7 \to Y, \, 8 \to Z, \\
9 &\to A, \, 10 \to B, \, 11 \to C, \, 12 \to D, \, 13 \to E, \, 14 \to F, \, 15 \to G, \, 16 \to H, \\
17 &\to I, \, 18 \to J, \, 19 \to K, \, 20 \to L.
\end{aligned}
}
\]