Factoring quadratic expressions worksheet with a decorative geometric design.

Let’s go step by step to factor each quadratic expression. We’re looking for two binomials that multiply to give the original trinomial (or difference of squares).

We’ll use these patterns:
- Perfect square trinomial: \( x^2 + 2ab + b^2 = (x + a)^2 \) or \( x^2 - 2ab + b^2 = (x - a)^2 \)
- Difference of squares: \( x^2 - a^2 = (x - a)(x + a) \)
- General trinomial: Find two numbers that multiply to the constant term and add to the middle coefficient.

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1. \( x^2 - 4 \)
This is difference of squares: \( x^2 - 2^2 = (x - 2)(x + 2) \) ✔

2. \( x^2 + 10x + 25 \)
Perfect square: \( (x + 5)^2 \) because 5×5=25 and 5+5=10 ✔

3. \( x^2 - 6x + 9 \)
Perfect square: \( (x - 3)^2 \) because (-3)×(-3)=9 and (-3)+(-3)=-6 ✔

4. \( x^2 + 16x + 64 \)
Perfect square: \( (x + 8)^2 \) because 8×8=64 and 8+8=16 ✔

5. \( x^2 - 36 \)
Difference of squares: \( (x - 6)(x + 6) \) ✔

6. \( x^2 - 2x + 1 \)
Perfect square: \( (x - 1)^2 \) ✔

7. \( x^2 + 6x + 9 \)
Perfect square: \( (x + 3)^2 \) ✔

8. \( x^2 - 25 \)
Difference of squares: \( (x - 5)(x + 5) \) ✔

9. \( x^2 - 81 \)
Difference of squares: \( (x - 9)(x + 9) \) ✔

10. \( x^2 - 18x + 81 \)
Perfect square: \( (x - 9)^2 \) ✔

11. \( x^2 + 14x + 49 \)
Perfect square: \( (x + 7)^2 \) ✔

12. \( x^2 - 16 \)
Difference of squares: \( (x - 4)(x + 4) \) ✔

13. \( x^2 + 2x + 1 \)
Perfect square: \( (x + 1)^2 \) ✔

Now the right column:

14. \( x^2 + 18x + 81 \)
Perfect square: \( (x + 9)^2 \) ✔

15. \( x^2 - 4x + 4 \)
Perfect square: \( (x - 2)^2 \) ✔

16. \( x^2 - 49 \)
Difference of squares: \( (x - 7)(x + 7) \) ✔

17. \( x^2 - 16x + 64 \)
Perfect square: \( (x - 8)^2 \) ✔

18. \( x^2 - 9 \)
Difference of squares: \( (x - 3)(x + 3) \) ✔

19. \( x^2 - 64 \)
Difference of squares: \( (x - 8)(x + 8) \) ✔

20. \( x^2 + 12x + 36 \)
Perfect square: \( (x + 6)^2 \) ✔

21. \( x^2 - 10x + 25 \)
Perfect square: \( (x - 5)^2 \) ✔

22. \( x^2 - 1 \)
Difference of squares: \( (x - 1)(x + 1) \) ✔

23. \( x^2 - 12x + 36 \)
Perfect square: \( (x - 6)^2 \) ✔

24. \( x^2 + 8x + 16 \)
Perfect square: \( (x + 4)^2 \) ✔

25. \( x^2 - 14x + 49 \)
Perfect square: \( (x - 7)^2 \) ✔

All factored expressions match the ones shown in the image — so they are all correct as given.

But wait — the task says “Factoring Quadratic Expressions” and shows already-factored forms next to them. The student may be expected to verify or just recognize the pattern. Since all are correctly factored, no correction needed.

However, if this is a matching or coloring activity (based on the grid), we don’t need to solve further — but since the user asked to “solve the problem accurately”, and all expressions are already factored correctly, the answer is that all factorizations shown are correct.

But perhaps the actual task is to *fill in* the factored form? Looking again — the left side has unfactored quadratics with blank boxes, and the right side has some filled in? Actually, re-examining: the image shows both the quadratic AND its factored form already written beside it — like an answer key.

Wait — maybe the student is supposed to match the factored form to the quadratic? But they’re already paired.

Alternatively, perhaps the black-and-white grid is to be colored based on which factored form matches — but without instructions, we can’t do that.

Given the context, I think the intended task is simply to confirm or write the factored form for each quadratic — and since they’re all already provided correctly, we can list them as answers.

But let me double-check one that might be tricky:

Look at #19: \( x^2 - 9 \) → should be \( (x - 3)(x + 3) \) — yes, correct.

#25: \( x^2 - 14x + 49 \) → \( (x - 7)^2 \) — yes.

All good.

So final answer: All expressions are correctly factored as shown.

But since the user likely wants the factored forms listed (even though they’re visible), I’ll provide them clearly.

Actually — looking back at the image description: the left column has quadratics with empty diamond boxes, and the right column has quadratics with filled-in factored forms? No — actually, in the image, every quadratic has its factored form written below it — even the ones on the left. For example:

Left top:
\( x^2 - 4 \)
\( (x - 2)(x + 2) \) ← already there

So perhaps the task is completed? Or maybe the student is to check their work?

Since the instruction is to “solve the problem accurately”, and all are already solved correctly, I will state that.

But to follow format strictly — perhaps the “problem” is to factor them, so I’ll output the factored forms as the answer.

Let me compile the full list of factored forms corresponding to each quadratic in order (left column first, then right column):

Left Column (top to bottom):

1. \( x^2 - 4 \) → \( (x - 2)(x + 2) \)

2. \( x^2 + 10x + 25 \) → \( (x + 5)^2 \)

3. \( x^2 - 6x + 9 \) → \( (x - 3)^2 \)

4. \( x^2 + 16x + 64 \) → \( (x + 8)^2 \)

5. \( x^2 - 36 \) → \( (x - 6)(x + 6) \)

6. \( x^2 - 2x + 1 \) → \( (x - 1)^2 \)

7. \( x^2 + 6x + 9 \) → \( (x + 3)^2 \)

8. \( x^2 - 25 \) → \( (x - 5)(x + 5) \)

9. \( x^2 - 81 \) → \( (x - 9)(x + 9) \)

10. \( x^2 - 18x + 81 \) → \( (x - 9)^2 \)

11. \( x^2 + 14x + 49 \) → \( (x + 7)^2 \)

12. \( x^2 - 16 \) → \( (x - 4)(x + 4) \)

13. \( x^2 + 2x + 1 \) → \( (x + 1)^2 \)

Right Column (top to bottom):

14. \( x^2 + 18x + 81 \) → \( (x + 9)^2 \)

15. \( x^2 - 4x + 4 \) → \( (x - 2)^2 \)

16. \( x^2 - 49 \) → \( (x - 7)(x + 7) \)

17. \( x^2 - 16x + 64 \) → \( (x - 8)^2 \)

18. \( x^2 - 9 \) → \( (x - 3)(x + 3) \)

19. \( x^2 - 64 \) → \( (x - 8)(x + 8) \)

20. \( x^2 + 12x + 36 \) → \( (x + 6)^2 \)

21. \( x^2 - 10x + 25 \) → \( (x - 5)^2 \)

22. \( x^2 - 1 \) → \( (x - 1)(x + 1) \)

23. \( x^2 - 12x + 36 \) → \( (x - 6)^2 \)

24. \( x^2 + 8x + 16 \) → \( (x + 4)^2 \)

25. \( x^2 - 14x + 49 \) → \( (x - 7)^2 \)

All match what’s shown. So the solution is verified.

Final Answer: All quadratic expressions are correctly factored as displayed in the worksheet.