Factorising quadratic equations worksheet with answers.

Problem Overview:

The task involves solving quadratic equations by factorizing them. The goal is to express each quadratic equation in the form of a product of two binomials and then solve for the variable.

General Approach:

1. Identify the quadratic equation: Each equation is in the standard form \( ax^2 + bx + c = 0 \).
2. Factorize the quadratic expression: Find two numbers that multiply to give \( ac \) (the product of the coefficient of \( x^2 \) and the constant term) and add up to \( b \) (the coefficient of \( x \)).
3. Rewrite the quadratic expression: Use the two numbers found to split the middle term and then factor by grouping.
4. Solve for the variable: Set each factor equal to zero and solve for the variable.

Detailed Solutions:

#### Problem 1: \( x^2 + 3x + 2 = 0 \)
- Factorize: \( x^2 + 3x + 2 = (x + 2)(x + 1) \)
- Solve:
\[
(x + 2)(x + 1) = 0 \implies x + 2 = 0 \text{ or } x + 1 = 0 \implies x = -2 \text{ or } x = -1
\]
- Answer: \( x = -2 \) or \( x = -1 \)

#### Problem 2: \( a^2 + 7a + 6 = 0 \)
- Factorize: \( a^2 + 7a + 6 = (a + 6)(a + 1) \)
- Solve:
\[
(a + 6)(a + 1) = 0 \implies a + 6 = 0 \text{ or } a + 1 = 0 \implies a = -6 \text{ or } a = -1
\]
- Answer: \( a = -6 \) or \( a = -1 \)

#### Problem 3: \( b^2 + 4b + 4 = 0 \)
- Factorize: \( b^2 + 4b + 4 = (b + 2)(b + 2) = (b + 2)^2 \)
- Solve:
\[
(b + 2)^2 = 0 \implies b + 2 = 0 \implies b = -2
\]
- Answer: \( b = -2 \)

#### Problem 4: \( c^2 + 5c + 6 = 0 \)
- Factorize: \( c^2 + 5c + 6 = (c + 3)(c + 2) \)
- Solve:
\[
(c + 3)(c + 2) = 0 \implies c + 3 = 0 \text{ or } c + 2 = 0 \implies c = -3 \text{ or } c = -2
\]
- Answer: \( c = -3 \) or \( c = -2 \)

#### Problem 5: \( d^2 + 8d + 12 = 0 \)
- Factorize: \( d^2 + 8d + 12 = (d + 6)(d + 2) \)
- Solve:
\[
(d + 6)(d + 2) = 0 \implies d + 6 = 0 \text{ or } d + 2 = 0 \implies d = -6 \text{ or } d = -2
\]
- Answer: \( d = -6 \) or \( d = -2 \)

#### Problem 6: \( e^2 + 6e + 8 = 0 \)
- Factorize: \( e^2 + 6e + 8 = (e + 4)(e + 2) \)
- Solve:
\[
(e + 4)(e + 2) = 0 \implies e + 4 = 0 \text{ or } e + 2 = 0 \implies e = -4 \text{ or } e = -2
\]
- Answer: \( e = -4 \) or \( e = -2 \)

#### Problem 7: \( f^2 + 7f + 10 = 0 \)
- Factorize: \( f^2 + 7f + 10 = (f + 5)(f + 2) \)
- Solve:
\[
(f + 5)(f + 2) = 0 \implies f + 5 = 0 \text{ or } f + 2 = 0 \implies f = -5 \text{ or } f = -2
\]
- Answer: \( f = -5 \) or \( f = -2 \)

#### Problem 8: \( g^2 + 8g + 7 = 0 \)
- Factorize: \( g^2 + 8g + 7 = (g + 7)(g + 1) \)
- Solve:
\[
(g + 7)(g + 1) = 0 \implies g + 7 = 0 \text{ or } g + 1 = 0 \implies g = -7 \text{ or } g = -1
\]
- Answer: \( g = -7 \) or \( g = -1 \)

#### Problem 9: \( h^2 + 10h + 9 = 0 \)
- Factorize: \( h^2 + 10h + 9 = (h + 9)(h + 1) \)
- Solve:
\[
(h + 9)(h + 1) = 0 \implies h + 9 = 0 \text{ or } h + 1 = 0 \implies h = -9 \text{ or } h = -1
\]
- Answer: \( h = -9 \) or \( h = -1 \)

#### Problem 10: \( i^2 + 8i + 16 = 0 \)
- Factorize: \( i^2 + 8i + 16 = (i + 4)(i + 4) = (i + 4)^2 \)
- Solve:
\[
(i + 4)^2 = 0 \implies i + 4 = 0 \implies i = -4
\]
- Answer: \( i = -4 \)

#### Problem 11: \( j^2 + 7j + 12 = 0 \)
- Factorize: \( j^2 + 7j + 12 = (j + 4)(j + 3) \)
- Solve:
\[
(j + 4)(j + 3) = 0 \implies j + 4 = 0 \text{ or } j + 3 = 0 \implies j = -4 \text{ or } j = -3
\]
- Answer: \( j = -4 \) or \( j = -3 \)

#### Problem 12: \( k^2 + 10k + 16 = 0 \)
- Factorize: \( k^2 + 10k + 16 = (k + 8)(k + 2) \)
- Solve:
\[
(k + 8)(k + 2) = 0 \implies k + 8 = 0 \text{ or } k + 2 = 0 \implies k = -8 \text{ or } k = -2
\]
- Answer: \( k = -8 \) or \( k = -2 \)

#### Problem 13: \( m^2 + 8m + 15 = 0 \)
- Factorize: \( m^2 + 8m + 15 = (m + 5)(m + 3) \)
- Solve:
\[
(m + 5)(m + 3) = 0 \implies m + 5 = 0 \text{ or } m + 3 = 0 \implies m = -5 \text{ or } m = -3
\]
- Answer: \( m = -5 \) or \( m = -3 \)

#### Problem 14: \( n^2 + 12n + 20 = 0 \)
- Factorize: \( n^2 + 12n + 20 = (n + 10)(n + 2) \)
- Solve:
\[
(n + 10)(n + 2) = 0 \implies n + 10 = 0 \text{ or } n + 2 = 0 \implies n = -10 \text{ or } n = -2
\]
- Answer: \( n = -10 \) or \( n = -2 \)

#### Problem 15: \( p^2 + 9p + 18 = 0 \)
- Factorize: \( p^2 + 9p + 18 = (p + 6)(p + 3) \)
- Solve:
\[
(p + 6)(p + 3) = 0 \implies p + 6 = 0 \text{ or } p + 3 = 0 \implies p = -6 \text{ or } p = -3
\]
- Answer: \( p = -6 \) or \( p = -3 \)

#### Problem 16: \( q^2 + 12q + 32 = 0 \)
- Factorize: \( q^2 + 12q + 32 = (q + 8)(q + 4) \)
- Solve:
\[
(q + 8)(q + 4) = 0 \implies q + 8 = 0 \text{ or } q + 4 = 0 \implies q = -8 \text{ or } q = -4
\]
- Answer: \( q = -8 \) or \( q = -4 \)

Final Answer:

\[
\boxed{
\begin{array}{lll}
1. & x = -2 \text{ or } x = -1 \\
2. & a = -6 \text{ or } a = -1 \\
3. & b = -2 \\
4. & c = -3 \text{ or } c = -2 \\
5. & d = -6 \text{ or } d = -2 \\
6. & e = -4 \text{ or } e = -2 \\
7. & f = -5 \text{ or } f = -2 \\
8. & g = -7 \text{ or } g = -1 \\
9. & h = -9 \text{ or } h = -1 \\
10. & i = -4 \\
11. & j = -4 \text{ or } j = -3 \\
12. & k = -8 \text{ or } k = -2 \\
13. & m = -5 \text{ or } m = -3 \\
14. & n = -10 \text{ or } n = -2 \\
15. & p = -6 \text{ or } p = -3 \\
16. & q = -8 \text{ or } q = -4 \\
\end{array}
}
\]