Perfect Squares and Factoring Worksheet for 9th - 11th Grade ... - Free Printable

Let’s solve each problem step by step. We’re factoring polynomials — that means rewriting them as a product of simpler expressions, if possible. If it can’t be factored (over integers), we write “prime”.

We’ll go one by one:

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1. x² + 6x + 9

Look for two numbers that multiply to 9 and add to 6 → 3 and 3.

So:
(x + 3)(x + 3) or (x + 3)²

✔ Factored

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2. x² - 4x + 4

Two numbers that multiply to 4 and add to -4 → -2 and -2.

So:
(x - 2)(x - 2) or (x - 2)²

✔ Factored

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3. x² - 8x + 16

Multiply to 16, add to -8 → -4 and -4.

So:
(x - 4)(x - 4) or (x - 4)²

✔ Factored

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4. x² - 10x + 25

Multiply to 25, add to -10 → -5 and -5.

So:
(x - 5)(x - 5) or (x - 5)²

✔ Factored

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5. x² + 14x + 49

Multiply to 49, add to 14 → 7 and 7.

So:
(x + 7)(x + 7) or (x + 7)²

✔ Factored

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6. x² - 12x + 36

Multiply to 36, add to -12 → -6 and -6.

So:
(x - 6)(x - 6) or (x - 6)²

✔ Factored

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7. x² - 18x + 81

Multiply to 81, add to -18 → -9 and -9.

So:
(x - 9)(x - 9) or (x - 9)²

✔ Factored

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8. x² - 20x + 100

Multiply to 100, add to -20 → -10 and -10.

So:
(x - 10)(x - 10) or (x - 10)²

✔ Factored

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9. 25x² - 30x + 9

This is not monic (leading coefficient ≠ 1). Check if it’s a perfect square trinomial.

√(25x²) = 5x
√9 = 3
Middle term should be 2*(5x)*(3) = 30x → but here it’s -30x → so yes!

So:
(5x - 3)(5x - 3) or (5x - 3)²

✔ Factored

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10. 16x² - 24x + 9

√(16x²) = 4x
√9 = 3
2*(4x)*(3) = 24x → matches middle term with negative sign.

So:
(4x - 3)(4x - 3) or (4x - 3)²

✔ Factored

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11. 4x² + 28x + 49

√(4x²) = 2x
√49 = 7
2*(2x)*(7) = 28x → matches!

So:
(2x + 7)(2x + 7) or (2x + 7)²

✔ Factored

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12. 9x² - 30x + 25

√(9x²) = 3x
√25 = 5
2*(3x)*(5) = 30x → matches with negative.

So:
(3x - 5)(3x - 5) or (3x - 5)²

✔ Factored

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13. 16x² - 56xy + 49y²

Treat like quadratic in x, with y as constant.

√(16x²) = 4x
√(49y²) = 7y
2*(4x)*(7y) = 56xy → matches with negative.

So:
(4x - 7y)(4x - 7y) or (4x - 7y)²

✔ Factored

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14. 25x² + 70xy + 49y²

√(25x²) = 5x
√(49y²) = 7y
2*(5x)*(7y) = 70xy → matches positive.

So:
(5x + 7y)(5x + 7y) or (5x + 7y)²

✔ Factored

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15. 4x² - 44xy + 121y²

√(4x²) = 2x
√(121y²) = 11y
2*(2x)*(11y) = 44xy → matches with negative.

So:
(2x - 11y)(2x - 11y) or (2x - 11y)²

✔ Factored

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16. 9x² - 60xy + 100y²

√(9x²) = 3x
√(100y²) = 10y
2*(3x)*(10y) = 60xy → matches with negative.

So:
(3x - 10y)(3x - 10y) or (3x - 10y)²

✔ Factored

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17. 25x² - 20xy + 4y²

√(25x²) = 5x
√(4y²) = 2y
2*(5x)*(2y) = 20xy → matches with negative.

So:
(5x - 2y)(5x - 2y) or (5x - 2y)²

✔ Factored

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18. 16x² - 8xy + y²

√(16x²) = 4x
√(y²) = y
2*(4x)*(y) = 8xy → matches with negative.

So:
(4x - y)(4x - y) or (4x - y)²

✔ Factored

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19. (1/4)x² - xy + y²

Fractions? Let’s factor out 1/4 first? Or treat as is.

Check: √((1/4)x²) = (1/2)x
√(y²) = y
2*(1/2 x)*(y) = xy → matches with negative.

So:
((1/2)x - y)((1/2)x - y) or ((1/2)x - y)²

But maybe better to write without fractions inside? Multiply through by 4 to check:

Original: (1/4)x² - xy + y²
Multiply by 4: x² - 4xy + 4y² = (x - 2y)²
So original = (1/4)(x - 2y)² = ((1/2)(x - 2y))² = ((1/2)x - y)² → same thing.

So either form is acceptable, but since the problem says “factor”, and this is a perfect square, we leave as:

((1/2)x - y)²

Alternatively, some might prefer to write as (x/2 - y)²

✔ Factored

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20. (1/9)x² - (2/3)xy + y²

Again, check perfect square.

√((1/9)x²) = (1/3)x
√(y²) = y
2*(1/3 x)*(y) = (2/3)xy → matches with negative.

So:
((1/3)x - y)((1/3)x - y) or ((1/3)x - y)²

Or equivalently: (x/3 - y)²

✔ Factored

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Now, let’s double-check if any were prime — all turned out to be perfect squares! So none are prime.

Wait — did I miss any? Let me scan again.

All 20 problems are perfect square trinomials. That makes sense — the worksheet title is “Perfect Squares and Factoring”.

So final answers:

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Final Answer:
1. (x + 3)²
2. (x - 2)²
3. (x - 4)²
4. (x - 5)²
5. (x + 7)²
6. (x - 6)²
7. (x - 9)²
8. (x - 10)²
9. (5x - 3)²
10. (4x - 3)²
11. (2x + 7)²
12. (3x - 5)²
13. (4x - 7y)²
14. (5x + 7y)²
15. (2x - 11y)²
16. (3x - 10y)²
17. (5x - 2y)²
18. (4x - y)²
19. ((1/2)x - y)² or (x/2 - y)²
20. ((1/3)x - y)² or (x/3 - y)²

Note: For #19 and #20, both forms are correct. Some teachers may prefer writing without fractions inside parentheses, but mathematically equivalent. Since the problem didn't specify, either is fine. I'll use the fractional coefficient form as written above.

Actually, let me verify #19 and #20 once more to be safe.

#19: (1/4)x² - xy + y²
= [ (1/2)x ]² - 2*(1/2 x)*y + y² → yes, perfect square → ((1/2)x - y)² ✔️

#20: (1/9)x² - (2/3)xy + y²
= [ (1/3)x ]² - 2*(1/3 x)*y + y² → yes → ((1/3)x - y)² ✔️

All good.

Final Answer:
1. (x + 3)²
2. (x - 2)²
3. (x - 4)²
4. (x - 5)²
5. (x + 7)²
6. (x - 6)²
7. (x - 9)²
8. (x - 10)²
9. (5x - 3)²
10. (4x - 3)²
11. (2x + 7)²
12. (3x - 5)²
13. (4x - 7y)²
14. (5x + 7y)²
15. (2x - 11y)²
16. (3x - 10y)²
17. (5x - 2y)²
18. (4x - y)²
19. \left(\frac{1}{2}x - y\right)^2
20. \left(\frac{1}{3}x - y\right)^2