Solving Quadratic Equations with Positive a Coefficients of 1 (A) - Free Printable

Let's solve each quadratic equation in the worksheet step by step. The general form of a quadratic equation is:

$$
ax^2 + bx + c = 0
$$

We can solve these equations using factoring, completing the square, or the quadratic formula:

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

But for most of these, factoring will work well.

---

1. $ x^2 + 5x + 6 = 0 $

Factor:
Find two numbers that multiply to 6 and add to 5 → 2 and 3
$$
(x + 2)(x + 3) = 0 \Rightarrow x = -2, -3
$$

---

2. $ x^2 - 36 = 0 $

This is a difference of squares:
$$
x^2 - 6^2 = (x - 6)(x + 6) = 0 \Rightarrow x = 6, -6
$$

---

3. $ x^2 - 2x - 15 = 0 $

Find two numbers that multiply to -15 and add to -2 → -5 and 3
$$
(x - 5)(x + 3) = 0 \Rightarrow x = 5, -3
$$

---

4. $ x^2 - 3x - 28 = 0 $

Multiply to -28, add to -3 → -7 and 4
$$
(x - 7)(x + 4) = 0 \Rightarrow x = 7, -4
$$

---

5. $ x^2 + 11x + 18 = 0 $

Multiply to 18, add to 11 → 9 and 2
$$
(x + 9)(x + 2) = 0 \Rightarrow x = -9, -2
$$

---

6. $ x^2 + 14x + 48 = 0 $

Multiply to 48, add to 14 → 6 and 8
$$
(x + 6)(x + 8) = 0 \Rightarrow x = -6, -8
$$

---

7. $ x^2 + x - 2 = 0 $

Multiply to -2, add to 1 → 2 and -1
$$
(x + 2)(x - 1) = 0 \Rightarrow x = -2, 1
$$

---

8. $ x^2 - 11x + 28 = 0 $

Multiply to 28, add to -11 → -7 and -4
$$
(x - 7)(x - 4) = 0 \Rightarrow x = 7, 4
$$

---

9. $ x^2 - 6x - 27 = 0 $

Multiply to -27, add to -6 → -9 and 3
$$
(x - 9)(x + 3) = 0 \Rightarrow x = 9, -3
$$

---

10. $ x^2 - 3x - 40 = 0 $

Multiply to -40, add to -3 → -8 and 5
$$
(x - 8)(x + 5) = 0 \Rightarrow x = 8, -5
$$

---

11. $ x^2 - 2x + 1 = 0 $

Perfect square trinomial:
$$
(x - 1)^2 = 0 \Rightarrow x = 1 \text{ (double root)}
$$

---

12. $ x^2 - 7x + 6 = 0 $

Multiply to 6, add to -7 → -6 and -1
$$
(x - 6)(x - 1) = 0 \Rightarrow x = 6, 1
$$

---

13. $ x^2 - 2x - 35 = 0 $

Multiply to -35, add to -2 → -7 and 5
$$
(x - 7)(x + 5) = 0 \Rightarrow x = 7, -5
$$

---

14. $ x^2 + 5x - 14 = 0 $

Multiply to -14, add to 5 → 7 and -2
$$
(x + 7)(x - 2) = 0 \Rightarrow x = -7, 2
$$

---

15. $ x^2 - x - 30 = 0 $

Multiply to -30, add to -1 → -6 and 5
$$
(x - 6)(x + 5) = 0 \Rightarrow x = 6, -5
$$

---

16. $ x^2 - 14x + 45 = 0 $

Multiply to 45, add to -14 → -9 and -5
$$
(x - 9)(x - 5) = 0 \Rightarrow x = 9, 5
$$

---

17. $ x^2 + 9x + 18 = 0 $

Multiply to 18, add to 9 → 6 and 3
$$
(x + 6)(x + 3) = 0 \Rightarrow x = -6, -3
$$

---

18. $ x^2 - 14x + 48 = 0 $

Multiply to 48, add to -14 → -8 and -6
$$
(x - 8)(x - 6) = 0 \Rightarrow x = 8, 6
$$

---

19. $ x^2 + 12x + 27 = 0 $

Multiply to 27, add to 12 → 9 and 3
$$
(x + 9)(x + 3) = 0 \Rightarrow x = -9, -3
$$

---

20. $ x^2 + 2x - 3 = 0 $

Multiply to -3, add to 2 → 3 and -1
$$
(x + 3)(x - 1) = 0 \Rightarrow x = -3, 1
$$

---

✔ Final Answers:

| Problem | Solutions |
|--------|-----------|
| 1 | $ x = -2, -3 $ |
| 2 | $ x = 6, -6 $ |
| 3 | $ x = 5, -3 $ |
| 4 | $ x = 7, -4 $ |
| 5 | $ x = -9, -2 $ |
| 6 | $ x = -6, -8 $ |
| 7 | $ x = -2, 1 $ |
| 8 | $ x = 7, 4 $ |
| 9 | $ x = 9, -3 $ |
|10 | $ x = 8, -5 $ |
|11 | $ x = 1 $ (double root) |
|12 | $ x = 6, 1 $ |
|13 | $ x = 7, -5 $ |
|14 | $ x = -7, 2 $ |
|15 | $ x = 6, -5 $ |
|16 | $ x = 9, 5 $ |
|17 | $ x = -6, -3 $ |
|18 | $ x = 8, 6 $ |
|19 | $ x = -9, -3 $ |
|20 | $ x = -3, 1 $ |

---

📝 Summary:

All equations were solved by factoring. Each was checked for patterns like:
- Difference of squares
- Perfect square trinomials
- Standard factoring with two binomials

If any equation didn't factor nicely, we'd use the quadratic formula — but all of these did!

Let me know if you want a PDF version or explanation of any specific one!