Factorising Quadratics 1 Textbook Exercise - Corbettmaths - Free Printable

Problem Overview:

The task involves factorizing quadratic expressions of the form \( ax^2 + bx + c \). For each quadratic expression, we need to find two binomials whose product equals the given quadratic. The general approach is to find two numbers that multiply to \( c \) (the constant term) and add up to \( b \) (the coefficient of \( x \)).

Solution:

#### Question 1: Factorize each of the following

1. (a) \( x^2 + 7x + 12 \)
- We need two numbers that multiply to \( 12 \) and add up to \( 7 \).
- The numbers are \( 3 \) and \( 4 \) because \( 3 \times 4 = 12 \) and \( 3 + 4 = 7 \).
- Therefore, \( x^2 + 7x + 12 = (x + 3)(x + 4) \).

2. (b) \( x^2 + 6x + 8 \)
- We need two numbers that multiply to \( 8 \) and add up to \( 6 \).
- The numbers are \( 2 \) and \( 4 \) because \( 2 \times 4 = 8 \) and \( 2 + 4 = 6 \).
- Therefore, \( x^2 + 6x + 8 = (x + 2)(x + 4) \).

3. (c) \( x^2 + 5x + 6 \)
- We need two numbers that multiply to \( 6 \) and add up to \( 5 \).
- The numbers are \( 2 \) and \( 3 \) because \( 2 \times 3 = 6 \) and \( 2 + 3 = 5 \).
- Therefore, \( x^2 + 5x + 6 = (x + 2)(x + 3) \).

4. (d) \( x^2 + 8x + 7 \)
- We need two numbers that multiply to \( 7 \) and add up to \( 8 \).
- The numbers are \( 1 \) and \( 7 \) because \( 1 \times 7 = 7 \) and \( 1 + 7 = 8 \).
- Therefore, \( x^2 + 8x + 7 = (x + 1)(x + 7) \).

5. (e) \( x^2 + 4x + 4 \)
- We need two numbers that multiply to \( 4 \) and add up to \( 4 \).
- The numbers are \( 2 \) and \( 2 \) because \( 2 \times 2 = 4 \) and \( 2 + 2 = 4 \).
- Therefore, \( x^2 + 4x + 4 = (x + 2)^2 \).

6. (f) \( x^2 + 8x + 15 \)
- We need two numbers that multiply to \( 15 \) and add up to \( 8 \).
- The numbers are \( 3 \) and \( 5 \) because \( 3 \times 5 = 15 \) and \( 3 + 5 = 8 \).
- Therefore, \( x^2 + 8x + 15 = (x + 3)(x + 5) \).

7. (g) \( x^2 + 6x + 9 \)
- We need two numbers that multiply to \( 9 \) and add up to \( 6 \).
- The numbers are \( 3 \) and \( 3 \) because \( 3 \times 3 = 9 \) and \( 3 + 3 = 6 \).
- Therefore, \( x^2 + 6x + 9 = (x + 3)^2 \).

8. (h) \( x^2 + 11x + 28 \)
- We need two numbers that multiply to \( 28 \) and add up to \( 11 \).
- The numbers are \( 4 \) and \( 7 \) because \( 4 \times 7 = 28 \) and \( 4 + 7 = 11 \).
- Therefore, \( x^2 + 11x + 28 = (x + 4)(x + 7) \).

9. (i) \( x^2 + 10x + 25 \)
- We need two numbers that multiply to \( 25 \) and add up to \( 10 \).
- The numbers are \( 5 \) and \( 5 \) because \( 5 \times 5 = 25 \) and \( 5 + 5 = 10 \).
- Therefore, \( x^2 + 10x + 25 = (x + 5)^2 \).

10. (j) \( x^2 + 12x + 20 \)
- We need two numbers that multiply to \( 20 \) and add up to \( 12 \).
- The numbers are \( 2 \) and \( 10 \) because \( 2 \times 10 = 20 \) and \( 2 + 10 = 12 \).
- Therefore, \( x^2 + 12x + 20 = (x + 2)(x + 10) \).

11. (k) \( x^2 + 25x + 24 \)
- We need two numbers that multiply to \( 24 \) and add up to \( 25 \).
- The numbers are \( 1 \) and \( 24 \) because \( 1 \times 24 = 24 \) and \( 1 + 24 = 25 \).
- Therefore, \( x^2 + 25x + 24 = (x + 1)(x + 24) \).

12. (l) \( x^2 + 11x + 24 \)
- We need two numbers that multiply to \( 24 \) and add up to \( 11 \).
- The numbers are \( 3 \) and \( 8 \) because \( 3 \times 8 = 24 \) and \( 3 + 8 = 11 \).
- Therefore, \( x^2 + 11x + 24 = (x + 3)(x + 8) \).

13. (m) \( x^2 + 9x + 14 \)
- We need two numbers that multiply to \( 14 \) and add up to \( 9 \).
- The numbers are \( 2 \) and \( 7 \) because \( 2 \times 7 = 14 \) and \( 2 + 7 = 9 \).
- Therefore, \( x^2 + 9x + 14 = (x + 2)(x + 7) \).

14. (n) \( x^2 + 23x + 60 \)
- We need two numbers that multiply to \( 60 \) and add up to \( 23 \).
- The numbers are \( 3 \) and \( 20 \) because \( 3 \times 20 = 60 \) and \( 3 + 20 = 23 \).
- Therefore, \( x^2 + 23x + 60 = (x + 3)(x + 20) \).

15. (o) \( x^2 + 29x + 100 \)
- We need two numbers that multiply to \( 100 \) and add up to \( 29 \).
- The numbers are \( 4 \) and \( 25 \) because \( 4 \times 25 = 100 \) and \( 4 + 25 = 29 \).
- Therefore, \( x^2 + 29x + 100 = (x + 4)(x + 25) \).

16. (p) \( x^2 + 20x + 51 \)
- We need two numbers that multiply to \( 51 \) and add up to \( 20 \).
- The numbers are \( 3 \) and \( 17 \) because \( 3 \times 17 = 51 \) and \( 3 + 17 = 20 \).
- Therefore, \( x^2 + 20x + 51 = (x + 3)(x + 17) \).

#### Question 2: Factorize each of the following

1. (a) \( x^2 + x - 12 \)
- We need two numbers that multiply to \( -12 \) and add up to \( 1 \).
- The numbers are \( 4 \) and \( -3 \) because \( 4 \times (-3) = -12 \) and \( 4 + (-3) = 1 \).
- Therefore, \( x^2 + x - 12 = (x + 4)(x - 3) \).

2. (b) \( x^2 + 5x - 6 \)
- We need two numbers that multiply to \( -6 \) and add up to \( 5 \).
- The numbers are \( 6 \) and \( -1 \) because \( 6 \times (-1) = -6 \) and \( 6 + (-1) = 5 \).
- Therefore, \( x^2 + 5x - 6 = (x + 6)(x - 1) \).

3. (c) \( x^2 + 3x - 10 \)
- We need two numbers that multiply to \( -10 \) and add up to \( 3 \).
- The numbers are \( 5 \) and \( -2 \) because \( 5 \times (-2) = -10 \) and \( 5 + (-2) = 3 \).
- Therefore, \( x^2 + 3x - 10 = (x + 5)(x - 2) \).

4. (d) \( x^2 + 3x - 4 \)
- We need two numbers that multiply to \( -4 \) and add up to \( 3 \).
- The numbers are \( 4 \) and \( -1 \) because \( 4 \times (-1) = -4 \) and \( 4 + (-1) = 3 \).
- Therefore, \( x^2 + 3x - 4 = (x + 4)(x - 1) \).

5. (e) \( x^2 + 2x - 48 \)
- We need two numbers that multiply to \( -48 \) and add up to \( 2 \).
- The numbers are \( 8 \) and \( -6 \) because \( 8 \times (-6) = -48 \) and \( 8 + (-6) = 2 \).
- Therefore, \( x^2 + 2x - 48 = (x + 8)(x - 6) \).

6. (f) \( x^2 + 4x - 32 \)
- We need two numbers that multiply to \( -32 \) and add up to \( 4 \).
- The numbers are \( 8 \) and \( -4 \) because \( 8 \times (-4) = -32 \) and \( 8 + (-4) = 4 \).
- Therefore, \( x^2 + 4x - 32 = (x + 8)(x - 4) \).

7. (g) \( x^2 + 2x - 35 \)
- We need two numbers that multiply to \( -35 \) and add up to \( 2 \).
- The numbers are \( 7 \) and \( -5 \) because \( 7 \times (-5) = -35 \) and \( 7 + (-5) = 2 \).
- Therefore, \( x^2 + 2x - 35 = (x + 7)(x - 5) \).

8. (h) \( x^2 + 8x - 33 \)
- We need two numbers that multiply to \( -33 \) and add up to \( 8 \).
- The numbers are \( 11 \) and \( -3 \) because \( 11 \times (-3) = -33 \) and \( 11 + (-3) = 8 \).
- Therefore, \( x^2 + 8x - 33 = (x + 11)(x - 3) \).

Final Answer:

\[
\boxed{
\begin{array}{ll}
\text{Question 1:} & \begin{aligned}
& (a) (x + 3)(x + 4) \\
& (b) (x + 2)(x + 4) \\
& (c) (x + 2)(x + 3) \\
& (d) (x + 1)(x + 7) \\
& (e) (x + 2)^2 \\
& (f) (x + 3)(x + 5) \\
& (g) (x + 3)^2 \\
& (h) (x + 4)(x + 7) \\
& (i) (x + 5)^2 \\
& (j) (x + 2)(x + 10) \\
& (k) (x + 1)(x + 24) \\
& (l) (x + 3)(x + 8) \\
& (m) (x + 2)(x + 7) \\
& (n) (x + 3)(x + 20) \\
& (o) (x + 4)(x + 25) \\
& (p) (x + 3)(x + 17)
\end{aligned} \\
\text{Question 2:} & \begin{aligned}
& (a) (x + 4)(x - 3) \\
& (b) (x + 6)(x - 1) \\
& (c) (x + 5)(x - 2) \\
& (d) (x + 4)(x - 1) \\
& (e) (x + 8)(x - 6) \\
& (f) (x + 8)(x - 4) \\
& (g) (x + 7)(x - 5) \\
& (h) (x + 11)(x - 3)
\end{aligned}
\end{array}
}
\]