Algebra worksheet featuring 27 problems on factoring quadratic expressions.

It looks like you've uploaded an image containing a list of algebraic expressions, but no specific task or question was included. The expressions are numbered from 13 to 27 and appear to be quadratic equations or polynomial expressions.

Here’s the list of expressions as they appear:

13. $ 6x^2 + 18x $
14. $ x^2 + 12x + 42 $
15. $ 3x^2 + 20x + 6 $
16. $ 2x^2 - 12x $
17. $ 48x^2 - 4x^3 $
18. $ 12x^2 + 12x - 24 $
19. $ 2x^2 - 9x - 12 $
20. $ x^2 + 10x + 25 $
21. $ 81x^2 - 25 $
22. $ 14x^2 + 2x^3 $
23. $ x^2 - 27 $
24. $ x^2 + 14x + 49 $
25. $ 4x^2 - 100 $
26. $ 16x^2y^2 + 2xyz - 8xy^3 $
27. $ 3x^2 - 6x + 9 $

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Possible Tasks:

Since no explicit instruction was given, here are common types of problems associated with such expressions:

1. Factor each expression
2. Solve for $ x $ (if it's an equation)
3. Simplify or rewrite in standard form
4. Identify if it's a perfect square trinomial, difference of squares, etc.

Let me assume the task is: Factor each expression completely.

I’ll go through them one by one and factor where possible.

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Factoring Each Expression:

13. $ 6x^2 + 18x $
Factor out GCF:
$ = 6x(x + 3) $

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14. $ x^2 + 12x + 42 $
Check discriminant: $ b^2 - 4ac = 144 - 168 = -24 < 0 $ → No real factors
→ Does not factor over real numbers
(Prime)

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15. $ 3x^2 + 20x + 6 $
Try factoring:
Need two numbers that multiply to $ 3 \cdot 6 = 18 $, add to 20 → none
Discriminant: $ 400 - 72 = 328 $, not a perfect square → No rational factors
→ Prime over integers

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16. $ 2x^2 - 12x $
Factor out GCF:
$ = 2x(x - 6) $

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17. $ 48x^2 - 4x^3 $
Rewrite: $ -4x^3 + 48x^2 $
Factor out GCF:
$ = -4x^2(x - 12) $

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18. $ 12x^2 + 12x - 24 $
Factor out GCF: $ 12(x^2 + x - 2) $
Now factor: $ x^2 + x - 2 = (x + 2)(x - 1) $
So: $ = 12(x + 2)(x - 1) $

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19. $ 2x^2 - 9x - 12 $
Use AC method: $ a=2, b=-9, c=-12 $ → $ ac = -24 $
Find two numbers: $ -12 $ and $ 2 $: $ (-12)(2) = -24 $, $ -12 + 2 = -10 $ → no
Try $ -8 $ and $ 3 $: $ -8+3=-5 $ → no
Try $ 3 $ and $ -8 $: same
Try $ 4 $ and $ -6 $: $ -2 $ → no
Try $ -6 $ and $ 4 $: $ -2 $ → no
Discriminant: $ 81 + 96 = 177 $ → not a perfect square → No rational factors
→ Prime over integers

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20. $ x^2 + 10x + 25 $
Perfect square trinomial:
$ = (x + 5)^2 $

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21. $ 81x^2 - 25 $
Difference of squares:
$ = (9x)^2 - (5)^2 = (9x - 5)(9x + 5) $

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22. $ 14x^2 + 2x^3 $
Rewrite: $ 2x^3 + 14x^2 $
Factor out GCF: $ 2x^2(x + 7) $

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23. $ x^2 - 27 $
Difference of squares? $ x^2 - (\sqrt{27})^2 $, but $ \sqrt{27} $ not integer
But $ 27 = 9 \cdot 3 $, so:
$ = x^2 - 27 = (x - \sqrt{27})(x + \sqrt{27}) $ → irrational
Or write as: $ x^2 - 27 $ → Not factorable over integers, but can be written as:
$ = (x - 3\sqrt{3})(x + 3\sqrt{3}) $ → irrational
→ Prime over integers

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24. $ x^2 + 14x + 49 $
Perfect square: $ = (x + 7)^2 $

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25. $ 4x^2 - 100 $
Factor out GCF: $ 4(x^2 - 25) $
Then: $ x^2 - 25 = (x - 5)(x + 5) $
So: $ = 4(x - 5)(x + 5) $

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26. $ 16x^2y^2 + 2xyz - 8xy^3 $
Factor out GCF: all terms have $ 2xy $?
Check:
- $ 16x^2y^2 $: yes
- $ 2xyz $: yes
- $ 8xy^3 $: yes
GCF: $ 2xy $
So:
$ = 2xy(8xy + z - 4y^2) $
Now look at inside: $ 8xy - 4y^2 + z $ → not easily factorable
So final answer: $ = 2xy(8xy - 4y^2 + z) $

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27. $ 3x^2 - 6x + 9 $
Factor out GCF: $ 3(x^2 - 2x + 3) $
Now check $ x^2 - 2x + 3 $:
Discriminant: $ 4 - 12 = -8 $ → no real roots → No real factors
→ So: $ = 3(x^2 - 2x + 3) $

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Summary of Factored Forms:

| Number | Expression | Factored Form |
|--------|------------|----------------|
| 13 | $ 6x^2 + 18x $ | $ 6x(x + 3) $ |
| 14 | $ x^2 + 12x + 42 $ | Prime |
| 15 | $ 3x^2 + 20x + 6 $ | Prime |
| 16 | $ 2x^2 - 12x $ | $ 2x(x - 6) $ |
| 17 | $ 48x^2 - 4x^3 $ | $ -4x^2(x - 12) $ |
| 18 | $ 12x^2 + 12x - 24 $ | $ 12(x + 2)(x - 1) $ |
| 19 | $ 2x^2 - 9x - 12 $ | Prime |
| 20 | $ x^2 + 10x + 25 $ | $ (x + 5)^2 $ |
| 21 | $ 81x^2 - 25 $ | $ (9x - 5)(9x + 5) $ |
| 22 | $ 14x^2 + 2x^3 $ | $ 2x^2(x + 7) $ |
| 23 | $ x^2 - 27 $ | Prime over integers |
| 24 | $ x^2 + 14x + 49 $ | $ (x + 7)^2 $ |
| 25 | $ 4x^2 - 100 $ | $ 4(x - 5)(x + 5) $ |
| 26 | $ 16x^2y^2 + 2xyz - 8xy^3 $ | $ 2xy(8xy - 4y^2 + z) $ |
| 27 | $ 3x^2 - 6x + 9 $ | $ 3(x^2 - 2x + 3) $ |

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