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

The task is to solve each quadratic equation for \( x \). Quadratic equations are of the form \( ax^2 + bx + c = 0 \), and they can be solved using various methods, including factoring, completing the square, or using the quadratic formula:

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

Step-by-Step Solutions

#### 1. \( x^2 + 5x + 6 = 0 \)
Factorize:
\[
x^2 + 5x + 6 = (x + 2)(x + 3) = 0
\]
Set each factor to zero:
\[
x + 2 = 0 \quad \text{or} \quad x + 3 = 0
\]
\[
x = -2 \quad \text{or} \quad x = -3
\]
Solution: \( x = -2, -3 \)

#### 2. \( x^2 - 36 = 0 \)
Rewrite as a difference of squares:
\[
x^2 - 36 = (x - 6)(x + 6) = 0
\]
Set each factor to zero:
\[
x - 6 = 0 \quad \text{or} \quad x + 6 = 0
\]
\[
x = 6 \quad \text{or} \quad x = -6
\]
Solution: \( x = 6, -6 \)

#### 3. \( x^2 - 2x - 15 = 0 \)
Factorize:
\[
x^2 - 2x - 15 = (x - 5)(x + 3) = 0
\]
Set each factor to zero:
\[
x - 5 = 0 \quad \text{or} \quad x + 3 = 0
\]
\[
x = 5 \quad \text{or} \quad x = -3
\]
Solution: \( x = 5, -3 \)

#### 4. \( x^2 - 3x - 28 = 0 \)
Factorize:
\[
x^2 - 3x - 28 = (x - 7)(x + 4) = 0
\]
Set each factor to zero:
\[
x - 7 = 0 \quad \text{or} \quad x + 4 = 0
\]
\[
x = 7 \quad \text{or} \quad x = -4
\]
Solution: \( x = 7, -4 \)

#### 5. \( x^2 + 11x + 18 = 0 \)
Factorize:
\[
x^2 + 11x + 18 = (x + 2)(x + 9) = 0
\]
Set each factor to zero:
\[
x + 2 = 0 \quad \text{or} \quad x + 9 = 0
\]
\[
x = -2 \quad \text{or} \quad x = -9
\]
Solution: \( x = -2, -9 \)

#### 6. \( x^2 + 14x + 48 = 0 \)
Factorize:
\[
x^2 + 14x + 48 = (x + 6)(x + 8) = 0
\]
Set each factor to zero:
\[
x + 6 = 0 \quad \text{or} \quad x + 8 = 0
\]
\[
x = -6 \quad \text{or} \quad x = -8
\]
Solution: \( x = -6, -8 \)

#### 7. \( x^2 + x - 2 = 0 \)
Factorize:
\[
x^2 + x - 2 = (x + 2)(x - 1) = 0
\]
Set each factor to zero:
\[
x + 2 = 0 \quad \text{or} \quad x - 1 = 0
\]
\[
x = -2 \quad \text{or} \quad x = 1
\]
Solution: \( x = -2, 1 \)

#### 8. \( x^2 - 11x + 28 = 0 \)
Factorize:
\[
x^2 - 11x + 28 = (x - 4)(x - 7) = 0
\]
Set each factor to zero:
\[
x - 4 = 0 \quad \text{or} \quad x - 7 = 0
\]
\[
x = 4 \quad \text{or} \quad x = 7
\]
Solution: \( x = 4, 7 \)

#### 9. \( x^2 - 6x - 27 = 0 \)
Factorize:
\[
x^2 - 6x - 27 = (x - 9)(x + 3) = 0
\]
Set each factor to zero:
\[
x - 9 = 0 \quad \text{or} \quad x + 3 = 0
\]
\[
x = 9 \quad \text{or} \quad x = -3
\]
Solution: \( x = 9, -3 \)

#### 10. \( x^2 - 3x - 40 = 0 \)
Factorize:
\[
x^2 - 3x - 40 = (x - 8)(x + 5) = 0
\]
Set each factor to zero:
\[
x - 8 = 0 \quad \text{or} \quad x + 5 = 0
\]
\[
x = 8 \quad \text{or} \quad x = -5
\]
Solution: \( x = 8, -5 \)

#### 11. \( x^2 - 2x + 1 = 0 \)
Factorize:
\[
x^2 - 2x + 1 = (x - 1)^2 = 0
\]
Set the factor to zero:
\[
x - 1 = 0
\]
\[
x = 1
\]
Solution: \( x = 1 \)

#### 12. \( x^2 - 7x + 6 = 0 \)
Factorize:
\[
x^2 - 7x + 6 = (x - 1)(x - 6) = 0
\]
Set each factor to zero:
\[
x - 1 = 0 \quad \text{or} \quad x - 6 = 0
\]
\[
x = 1 \quad \text{or} \quad x = 6
\]
Solution: \( x = 1, 6 \)

#### 13. \( x^2 - 2x - 35 = 0 \)
Factorize:
\[
x^2 - 2x - 35 = (x - 7)(x + 5) = 0
\]
Set each factor to zero:
\[
x - 7 = 0 \quad \text{or} \quad x + 5 = 0
\]
\[
x = 7 \quad \text{or} \quad x = -5
\]
Solution: \( x = 7, -5 \)

#### 14. \( x^2 + 5x - 14 = 0 \)
Factorize:
\[
x^2 + 5x - 14 = (x + 7)(x - 2) = 0
\]
Set each factor to zero:
\[
x + 7 = 0 \quad \text{or} \quad x - 2 = 0
\]
\[
x = -7 \quad \text{or} \quad x = 2
\]
Solution: \( x = -7, 2 \)

#### 15. \( x^2 - x - 30 = 0 \)
Factorize:
\[
x^2 - x - 30 = (x - 6)(x + 5) = 0
\]
Set each factor to zero:
\[
x - 6 = 0 \quad \text{or} \quad x + 5 = 0
\]
\[
x = 6 \quad \text{or} \quad x = -5
\]
Solution: \( x = 6, -5 \)

#### 16. \( x^2 - 14x + 45 = 0 \)
Factorize:
\[
x^2 - 14x + 45 = (x - 5)(x - 9) = 0
\]
Set each factor to zero:
\[
x - 5 = 0 \quad \text{or} \quad x - 9 = 0
\]
\[
x = 5 \quad \text{or} \quad x = 9
\]
Solution: \( x = 5, 9 \)

#### 17. \( x^2 + 9x + 18 = 0 \)
Factorize:
\[
x^2 + 9x + 18 = (x + 3)(x + 6) = 0
\]
Set each factor to zero:
\[
x + 3 = 0 \quad \text{or} \quad x + 6 = 0
\]
\[
x = -3 \quad \text{or} \quad x = -6
\]
Solution: \( x = -3, -6 \)

#### 18. \( x^2 - 14x + 48 = 0 \)
Factorize:
\[
x^2 - 14x + 48 = (x - 6)(x - 8) = 0
\]
Set each factor to zero:
\[
x - 6 = 0 \quad \text{or} \quad x - 8 = 0
\]
\[
x = 6 \quad \text{or} \quad x = 8
\]
Solution: \( x = 6, 8 \)

#### 19. \( x^2 + 12x + 27 = 0 \)
Factorize:
\[
x^2 + 12x + 27 = (x + 3)(x + 9) = 0
\]
Set each factor to zero:
\[
x + 3 = 0 \quad \text{or} \quad x + 9 = 0
\]
\[
x = -3 \quad \text{or} \quad x = -9
\]
Solution: \( x = -3, -9 \)

#### 20. \( x^2 + 2x - 3 = 0 \)
Factorize:
\[
x^2 + 2x - 3 = (x + 3)(x - 1) = 0
\]
Set each factor to zero:
\[
x + 3 = 0 \quad \text{or} \quad x - 1 = 0
\]
\[
x = -3 \quad \text{or} \quad x = 1
\]
Solution: \( x = -3, 1 \)

Final Answer

\[
\boxed{
\begin{array}{ll}
1. & x = -2, -3 \\
2. & x = 6, -6 \\
3. & x = 5, -3 \\
4. & x = 7, -4 \\
5. & x = -2, -9 \\
6. & x = -6, -8 \\
7. & x = -2, 1 \\
8. & x = 4, 7 \\
9. & x = 9, -3 \\
10. & x = 8, -5 \\
11. & x = 1 \\
12. & x = 1, 6 \\
13. & x = 7, -5 \\
14. & x = -7, 2 \\
15. & x = 6, -5 \\
16. & x = 5, 9 \\
17. & x = -3, -6 \\
18. & x = 6, 8 \\
19. & x = -3, -9 \\
20. & x = -3, 1 \\
\end{array}
}
\]