Factoring Quadratic Equations Worksheets - Free Printable

Explanation:
We are asked to factorise each quadratic equation, using positive values only — meaning we want factors with positive numbers (e.g., $(x+2)(x+1)$, not $(x-2)(x-1)$), and the solutions (roots) should be negative numbers (since $x = -a$ or $-b$ when factoring as $(x+a)(x+b)=0$).

Each quadratic is of the form:
$$
\text{variable}^2 + (\text{sum})\cdot\text{variable} + (\text{product}) = 0
$$
We need two numbers that:
- Multiply to give the constant term (the last number),
- Add to give the coefficient of the middle term.

Let’s go one by one:

---

2) $a^2 + 7a + 6 = 0$
Find two numbers that multiply to $+6$, add to $+7$:
→ $1$ and $6$: $1 \times 6 = 6$, $1 + 6 = 7$
So: $(a + 1)(a + 6) = 0$
Roots: $a = -1$ or $-6$

3) $b^2 + 4b + 4 = 0$
Numbers multiplying to $4$, adding to $4$:
→ $2$ and $2$: $2 \times 2 = 4$, $2 + 2 = 4$
So: $(b + 2)(b + 2) = (b + 2)^2 = 0$
Root: $b = -2$ (double root)

4) $c^2 + 5c + 6 = 0$
Numbers: $2$ and $3$: $2 \times 3 = 6$, $2 + 3 = 5$
→ $(c + 2)(c + 3) = 0$
Roots: $c = -2$ or $-3$

5) $d^2 + 8d + 12 = 0$
Numbers: $2$ and $6$: $2 \times 6 = 12$, $2 + 6 = 8$
→ $(d + 2)(d + 6) = 0$
Roots: $d = -2$ or $-6$

6) $e^2 + 6e + 8 = 0$
Numbers: $2$ and $4$: $2 \times 4 = 8$, $2 + 4 = 6$
→ $(e + 2)(e + 4) = 0$
Roots: $e = -2$ or $-4$

7) $f^2 + 7f + 10 = 0$
Numbers: $2$ and $5$: $2 \times 5 = 10$, $2 + 5 = 7$
→ $(f + 2)(f + 5) = 0$
Roots: $f = -2$ or $-5$

8) $g^2 + 8g + 7 = 0$
Numbers: $1$ and $7$: $1 \times 7 = 7$, $1 + 7 = 8$
→ $(g + 1)(g + 7) = 0$
Roots: $g = -1$ or $-7$

9) $h^2 + 10h + 9 = 0$
Numbers: $1$ and $9$: $1 \times 9 = 9$, $1 + 9 = 10$
→ $(h + 1)(h + 9) = 0$
Roots: $h = -1$ or $-9$

10) $i^2 + 8i + 16 = 0$
Numbers: $4$ and $4$: $4 \times 4 = 16$, $4 + 4 = 8$
→ $(i + 4)^2 = 0$
Root: $i = -4$

11) $j^2 + 7j + 12 = 0$
Numbers: $3$ and $4$: $3 \times 4 = 12$, $3 + 4 = 7$
→ $(j + 3)(j + 4) = 0$
Roots: $j = -3$ or $-4$

12) $k^2 + 10k + 16 = 0$
Numbers: $2$ and $8$: $2 \times 8 = 16$, $2 + 8 = 10$
→ $(k + 2)(k + 8) = 0$
Roots: $k = -2$ or $-8$

13) $m^2 + 8m + 15 = 0$
Numbers: $3$ and $5$: $3 \times 5 = 15$, $3 + 5 = 8$
→ $(m + 3)(m + 5) = 0$
Roots: $m = -3$ or $-5$

14) $n^2 + 12n + 20 = 0$
Numbers: $2$ and $10$: $2 \times 10 = 20$, $2 + 10 = 12$
→ $(n + 2)(n + 10) = 0$
Roots: $n = -2$ or $-10$

15) $p^2 + 9p + 18 = 0$
Numbers: $3$ and $6$: $3 \times 6 = 18$, $3 + 6 = 9$
→ $(p + 3)(p + 6) = 0$
Roots: $p = -3$ or $-6$

16) $q^2 + 12q + 32 = 0$
Numbers: $4$ and $8$: $4 \times 8 = 32$, $4 + 8 = 12$
→ $(q + 4)(q + 8) = 0$
Roots: $q = -4$ or $-8$

Let me double-check a few to avoid mistakes:

- #12: $k^2 + 10k + 16$: $(k+2)(k+8) = k^2 + 8k + 2k + 16 = k^2 + 10k + 16$ ✔
- #16: $(q+4)(q+8) = q^2 + 8q + 4q + 32 = q^2 + 12q + 32$ ✔
- #10: $(i+4)^2 = i^2 + 8i + 16$ ✔

All correct.

Now, for the answer format, since the worksheet has three columns (equation, factorised form, roots), and only #1 is filled, we need to fill in the rest similarly.

But the instruction says: “Solve the problem accurately.” And the user wants only the final answer — likely the completed rows (factorised forms and roots) for questions 2–16.

Since the output must be only the Final Answer (after reasoning), I’ll list each row’s factorised form and solution, matching the format of row 1:

Row 1 format:
Equation | Factorised | Roots

So for each:

2) $a^2 + 7a + 6 = 0$ → $(a+1)(a+6)=0$ → $a = -1$ or $-6$
3) $b^2 + 4b + 4 = 0$ → $(b+2)^2 = 0$ → $b = -2$
4) $c^2 + 5c + 6 = 0$ → $(c+2)(c+3)=0$ → $c = -2$ or $-3$
5) $d^2 + 8d + 12 = 0$ → $(d+2)(d+6)=0$ → $d = -2$ or $-6$
6) $e^2 + 6e + 8 = 0$ → $(e+2)(e+4)=0$ → $e = -2$ or $-4$
7) $f^2 + 7f + 10 = 0$ → $(f+2)(f+5)=0$ → $f = -2$ or $-5$
8) $g^2 + 8g + 7 = 0$ → $(g+1)(g+7)=0$ → $g = -1$ or $-7$
9) $h^2 + 10h + 9 = 0$ → $(h+1)(h+9)=0$ → $h = -1$ or $-9$
10) $i^2 + 8i + 16 = 0$ → $(i+4)^2 = 0$ → $i = -4$
11) $j^2 + 7j + 12 = 0$ → $(j+3)(j+4)=0$ → $j = -3$ or $-4$
12) $k^2 + 10k + 16 = 0$ → $(k+2)(k+8)=0$ → $k = -2$ or $-8$
13) $m^2 + 8m + 15 = 0$ → $(m+3)(m+5)=0$ → $m = -3$ or $-5$
14) $n^2 + 12n + 20 = 0$ → $(n+2)(n+10)=0$ → $n = -2$ or $-10$
15) $p^2 + 9p + 18 = 0$ → $(p+3)(p+6)=0$ → $p = -3$ or $-6$
16) $q^2 + 12q + 32 = 0$ → $(q+4)(q+8)=0$ → $q = -4$ or $-8$

Final Answer:
2) $(a+1)(a+6)=0$, $a = -1$ or $-6$
3) $(b+2)^2=0$, $b = -2$
4) $(c+2)(c+3)=0$, $c = -2$ or $-3$
5) $(d+2)(d+6)=0$, $d = -2$ or $-6$
6) $(e+2)(e+4)=0$, $e = -2$ or $-4$
7) $(f+2)(f+5)=0$, $f = -2$ or $-5$
8) $(g+1)(g+7)=0$, $g = -1$ or $-7$
9) $(h+1)(h+9)=0$, $h = -1$ or $-9$
10) $(i+4)^2=0$, $i = -4$
11) $(j+3)(j+4)=0$, $j = -3$ or $-4$
12) $(k+2)(k+8)=0$, $k = -2$ or $-8$
13) $(m+3)(m+5)=0$, $m = -3$ or $-5$
14) $(n+2)(n+10)=0$, $n = -2$ or $-10$
15) $(p+3)(p+6)=0$, $p = -3$ or $-6$
16) $(q+4)(q+8)=0$, $q = -4$ or $-8$