Factoring Quadratics (a=1) differentiated worksheet with three levels of difficulty for students to practice factoring quadratic expressions.

The image shows a "Factoring Quadratics (a=1) Differentiated Worksheet" from Ten Tors Math, designed to help students practice factoring quadratic expressions where the leading coefficient is 1 (i.e., quadratics in the form $ x^2 + bx + c $).

There are three columns of problems, each with increasing difficulty indicated by crown icons:
- One crown: easiest
- Two crowns: medium
- Three crowns: hardest

Let’s solve each problem step-by-step and explain the method.

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🔷 Method for Factoring Quadratics (when $ a = 1 $)

For a quadratic expression like $ x^2 + bx + c $, we want to factor it into two binomials:
$$
x^2 + bx + c = (x + m)(x + n)
$$
where:
- $ m \times n = c $ (the constant term)
- $ m + n = b $ (the coefficient of the middle term)

We look for two numbers that multiply to $ c $ and add to $ b $.

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## ✔ Column 1 (One Crown - Easiest)

1) $ x^2 + 2x + 1 $

- Find two numbers that multiply to 1 and add to 2 → $ 1 $ and $ 1 $
- $ (x + 1)(x + 1) = (x + 1)^2 $

✔ Answer: $ (x + 1)^2 $

2) $ x^2 + 4x + 3 $

- Multiply to 3, add to 4 → $ 1 $ and $ 3 $
- $ (x + 1)(x + 3) $

✔ Answer: $ (x + 1)(x + 3) $

3) $ x^2 + 7x + 6 $

- Multiply to 6, add to 7 → $ 1 $ and $ 6 $
- $ (x + 1)(x + 6) $

✔ Answer: $ (x + 1)(x + 6) $

4) $ x^2 + 7x + 12 $

- Multiply to 12, add to 7 → $ 3 $ and $ 4 $
- $ (x + 3)(x + 4) $

✔ Answer: $ (x + 3)(x + 4) $

5) $ x^2 + 6x + 9 $

- Multiply to 9, add to 6 → $ 3 $ and $ 3 $
- $ (x + 3)^2 $

✔ Answer: $ (x + 3)^2 $

6) $ x^2 + 8x + 15 $

- Multiply to 15, add to 8 → $ 3 $ and $ 5 $
- $ (x + 3)(x + 5) $

✔ Answer: $ (x + 3)(x + 5) $

7) $ x^2 + 10x + 16 $

- Multiply to 16, add to 10 → $ 2 $ and $ 8 $
- $ (x + 2)(x + 8) $

✔ Answer: $ (x + 2)(x + 8) $

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## ✔ Column 2 (Two Crowns - Medium)

1) $ x^2 - 4x + 3 $

- Multiply to 3, add to -4 → $ -1 $ and $ -3 $
- $ (x - 1)(x - 3) $

✔ Answer: $ (x - 1)(x - 3) $

2) $ x^2 - 8x + 15 $

- Multiply to 15, add to -8 → $ -3 $ and $ -5 $
- $ (x - 3)(x - 5) $

✔ Answer: $ (x - 3)(x - 5) $

3) $ x^2 - 10x + 9 $

- Multiply to 9, add to -10 → $ -1 $ and $ -9 $
- $ (x - 1)(x - 9) $

✔ Answer: $ (x - 1)(x - 9) $

4) $ x^2 - 9x + 18 $

- Multiply to 18, add to -9 → $ -3 $ and $ -6 $
- $ (x - 3)(x - 6) $

✔ Answer: $ (x - 3)(x - 6) $

5) $ x^2 - 8x + 16 $

- Multiply to 16, add to -8 → $ -4 $ and $ -4 $
- $ (x - 4)^2 $

✔ Answer: $ (x - 4)^2 $

6) $ x^2 - 10x + 21 $

- Multiply to 21, add to -10 → $ -3 $ and $ -7 $
- $ (x - 3)(x - 7) $

✔ Answer: $ (x - 3)(x - 7) $

7) $ x^2 - 11x + 28 $

- Multiply to 28, add to -11 → $ -4 $ and $ -7 $
- $ (x - 4)(x - 7) $

✔ Answer: $ (x - 4)(x - 7) $

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## ✔ Column 3 (Three Crowns - Hardest)

These involve negative constants or mixed signs.

1) $ x^2 - 2x - 4 $

- Multiply to -4, add to -2 → Try pairs: $ 1 $ and $ -4 $ → sum = -3 ✘
$ -1 $ and $ 4 $ → sum = 3 ✘
$ 2 $ and $ -2 $ → sum = 0 ✘
No integer pair adds to -2 and multiplies to -4 → Not factorable over integers

✘ This one does not factor nicely (irreducible over integers).
But let's check again carefully:
- $ x^2 - 2x - 4 $ → discriminant $ = (-2)^2 - 4(1)(-4) = 4 + 16 = 20 $, not a perfect square → no rational factors

✔ Answer: Does not factor (or use quadratic formula)

> Note: Maybe typo? If it were $ x^2 - 2x - 3 $, then $ (x - 3)(x + 1) $. But as is, not factorable.

Wait — let’s recheck the original problem. It says:

> 1) $ x^2 - 2x - 4 $

Yes, this one cannot be factored using integers.

So answer: Not factorable (over integers)

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2) $ x^2 - 3x - 18 $

- Multiply to -18, add to -3 → $ -6 $ and $ 3 $ → sum = -3 ✔
- $ (x - 6)(x + 3) $

✔ Answer: $ (x - 6)(x + 3) $

3) $ x^2 + 8x - 20 $

- Multiply to -20, add to 8 → $ 10 $ and $ -2 $ → sum = 8 ✔
- $ (x + 10)(x - 2) $

✔ Answer: $ (x + 10)(x - 2) $

4) $ x^2 + x - 20 $

- Multiply to -20, add to 1 → $ 5 $ and $ -4 $ → sum = 1 ✔
- $ (x + 5)(x - 4) $

✔ Answer: $ (x + 5)(x - 4) $

5) $ x^2 - 9 $

- Difference of squares: $ x^2 - 3^2 = (x - 3)(x + 3) $

✔ Answer: $ (x - 3)(x + 3) $

6) $ x^2 + x - 42 $

- Multiply to -42, add to 1 → $ 7 $ and $ -6 $ → sum = 1 ✔
- $ (x + 7)(x - 6) $

✔ Answer: $ (x + 7)(x - 6) $

7) $ x^2 - x - 42 $

- Multiply to -42, add to -1 → $ -7 $ and $ 6 $ → sum = -1 ✔
- $ (x - 7)(x + 6) $

✔ Answer: $ (x - 7)(x + 6) $

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## ✔ Summary of Answers

Column 1:

1. $ (x + 1)^2 $
2. $ (x + 1)(x + 3) $
3. $ (x + 1)(x + 6) $
4. $ (x + 3)(x + 4) $
5. $ (x + 3)^2 $
6. $ (x + 3)(x + 5) $
7. $ (x + 2)(x + 8) $

Column 2:

1. $ (x - 1)(x - 3) $
2. $ (x - 3)(x - 5) $
3. $ (x - 1)(x - 9) $
4. $ (x - 3)(x - 6) $
5. $ (x - 4)^2 $
6. $ (x - 3)(x - 7) $
7. $ (x - 4)(x - 7) $

Column 3:

1. Not factorable (over integers)
2. $ (x - 6)(x + 3) $
3. $ (x + 10)(x - 2) $
4. $ (x + 5)(x - 4) $
5. $ (x - 3)(x + 3) $
6. $ (x + 7)(x - 6) $
7. $ (x - 7)(x + 6) $

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📝 Notes:

- The worksheet is differentiated: easier problems first, harder ones later.
- Some problems (like #1 in column 3) may not factor neatly — this teaches students to recognize when factoring isn’t possible with integers.
- Always check: Multiply the factors to verify they give back the original quadratic.

Let me know if you'd like a printable version or explanation for non-factorable cases!