Factoring Quadratic Equations Worksheets - Free Printable

Let's solve each quadratic equation by factorising. We are asked to factorise the quadratic equations and find the solutions (roots), but only positive values are required — however, since these are standard quadratic equations with real roots, we will find all solutions and then note which ones are positive.

We'll follow the method of splitting the middle term or using trial and error to find two numbers that:
- Multiply to give the constant term (the number without a variable),
- Add to give the coefficient of the middle term.

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1) $ x^2 + 3x + 2 = 0 $

We need two numbers that multiply to 2 and add to 3 → 2 and 1
So:
$ x^2 + 2x + x + 2 = (x+2)(x+1) = 0 $
Solutions: $ x = -2 $ or $ x = -1 $ → No positive solutions

✔ Factored: $ (x+2)(x+1) $
✘ No positive roots

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2) $ a^2 + 7a + 6 = 0 $

Numbers: 6 and 1 → $ 6 \times 1 = 6 $, $ 6 + 1 = 7 $
$ a^2 + 6a + a + 6 = (a+6)(a+1) = 0 $
Solutions: $ a = -6 $ or $ a = -1 $ → ✘ No positive

✔ $ (a+6)(a+1) $

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3) $ b^2 + 4b + 4 = 0 $

This is a perfect square: $ (b+2)^2 = 0 $
So $ b = -2 $ → ✘ Not positive

✔ $ (b+2)^2 $

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4) $ c^2 + 5c + 6 = 0 $

Need: 2 and 3 → $ 2 \times 3 = 6 $, $ 2 + 3 = 5 $
$ (c+2)(c+3) = 0 $ → $ c = -2 $ or $ c = -3 $ → ✘ No positive

✔ $ (c+2)(c+3) $

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5) $ d^2 + 8d + 12 = 0 $

Need: 6 and 2 → $ 6 \times 2 = 12 $, $ 6 + 2 = 8 $
$ (d+6)(d+2) = 0 $ → $ d = -6 $ or $ d = -2 $ → ✘ No positive

✔ $ (d+6)(d+2) $

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6) $ e^2 + 6e + 8 = 0 $

Need: 4 and 2 → $ 4 \times 2 = 8 $, $ 4 + 2 = 6 $
$ (e+4)(e+2) = 0 $ → $ e = -4 $ or $ e = -2 $ → ✘ No positive

✔ $ (e+4)(e+2) $

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7) $ f^2 + 7f + 10 = 0 $

Need: 5 and 2 → $ 5 \times 2 = 10 $, $ 5 + 2 = 7 $
$ (f+5)(f+2) = 0 $ → $ f = -5 $ or $ f = -2 $ → ✘ No positive

✔ $ (f+5)(f+2) $

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8) $ g^2 + 8g + 7 = 0 $

Need: 7 and 1 → $ 7 \times 1 = 7 $, $ 7 + 1 = 8 $
$ (g+7)(g+1) = 0 $ → $ g = -7 $ or $ g = -1 $ → ✘ No positive

✔ $ (g+7)(g+1) $

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9) $ h^2 + 10h + 9 = 0 $

Need: 9 and 1 → $ 9 \times 1 = 9 $, $ 9 + 1 = 10 $
$ (h+9)(h+1) = 0 $ → $ h = -9 $ or $ h = -1 $ → ✘ No positive

✔ $ (h+9)(h+1) $

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10) $ i^2 + 8i + 16 = 0 $

Perfect square: $ (i+4)^2 = 0 $ → $ i = -4 $ → ✘ Not positive

✔ $ (i+4)^2 $

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11) $ j^2 + 7j + 12 = 0 $

Need: 4 and 3 → $ 4 \times 3 = 12 $, $ 4 + 3 = 7 $
$ (j+4)(j+3) = 0 $ → $ j = -4 $ or $ j = -3 $ → ✘ No positive

✔ $ (j+4)(j+3) $

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12) $ k^2 + 10k + 16 = 0 $

Need: 8 and 2 → $ 8 \times 2 = 16 $, $ 8 + 2 = 10 $
$ (k+8)(k+2) = 0 $ → $ k = -8 $ or $ k = -2 $ → ✘ No positive

✔ $ (k+8)(k+2) $

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13) $ m^2 + 8m + 15 = 0 $

Need: 5 and 3 → $ 5 \times 3 = 15 $, $ 5 + 3 = 8 $
$ (m+5)(m+3) = 0 $ → $ m = -5 $ or $ m = -3 $ → ✘ No positive

✔ $ (m+5)(m+3) $

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14) $ n^2 + 12n + 20 = 0 $

Need: 10 and 2 → $ 10 \times 2 = 20 $, $ 10 + 2 = 12 $
$ (n+10)(n+2) = 0 $ → $ n = -10 $ or $ n = -2 $ → ✘ No positive

✔ $ (n+10)(n+2) $

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15) $ p^2 + 9p + 18 = 0 $

Need: 6 and 3 → $ 6 \times 3 = 18 $, $ 6 + 3 = 9 $
$ (p+6)(p+3) = 0 $ → $ p = -6 $ or $ p = -3 $ → ✘ No positive

✔ $ (p+6)(p+3) $

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16) $ q^2 + 12q + 32 = 0 $

Need: 8 and 4 → $ 8 \times 4 = 32 $, $ 8 + 4 = 12 $
$ (q+8)(q+4) = 0 $ → $ q = -8 $ or $ q = -4 $ → ✘ No positive

✔ $ (q+8)(q+4) $

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✔ Final Answers Summary:

| Problem | Factorised Form | Solutions |
|--------|------------------|-----------|
| 1) | $ (x+2)(x+1) $ | $ x = -2, -1 $ |
| 2) | $ (a+6)(a+1) $ | $ a = -6, -1 $ |
| 3) | $ (b+2)^2 $ | $ b = -2 $ |
| 4) | $ (c+2)(c+3) $ | $ c = -2, -3 $ |
| 5) | $ (d+6)(d+2) $ | $ d = -6, -2 $ |
| 6) | $ (e+4)(e+2) $ | $ e = -4, -2 $ |
| 7) | $ (f+5)(f+2) $ | $ f = -5, -2 $ |
| 8) | $ (g+7)(g+1) $ | $ g = -7, -1 $ |
| 9) | $ (h+9)(h+1) $ | $ h = -9, -1 $ |
| 10) | $ (i+4)^2 $ | $ i = -4 $ |
| 11) | $ (j+4)(j+3) $ | $ j = -4, -3 $ |
| 12) | $ (k+8)(k+2) $ | $ k = -8, -2 $ |
| 13) | $ (m+5)(m+3) $ | $ m = -5, -3 $ |
| 14) | $ (n+10)(n+2) $| $ n = -10, -2 $ |
| 15) | $ (p+6)(p+3) $ | $ p = -6, -3 $ |
| 16) | $ (q+8)(q+4) $ | $ q = -8, -4 $ |

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🚨 Important Note:

All equations have only negative solutions, so there are no positive values for any variable in this worksheet.

The instruction says "Positive values only", which might mean that you should only write down positive solutions if they exist — but in this case, none do.

So, for each problem, the factorisation is correct, but no positive solution exists.

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✔ Final Answer:

All quadratics factorise as shown above. However, none of them have positive solutions — all roots are negative.

Let me know if you'd like help modifying the equations to have positive roots!