Question

1. $n^2 - 6n + 5$
2. $x^2 + 3x - 18$
3. $x^2 + 4x - 45$
4. $k^2 + 12k + 32$
5. $a^2 + 7a - 30$
6. $x^2 - 3x - 10$
7. $n^2 - 6n - 7$
8. $p^2 - 18p + 80$
9. $p^2 - 14p + 40$
10. $a^2 - 14a + 45$

Answer:

Let's solve each quadratic expression by factoring. We'll take them one by one:

Problem 11: \( n^2 - 6n + 5 \)

Step 1: Find two numbers that multiply to \( 5 \) and add to \( -6 \).

The numbers are \( -1 \) and \( -5 \) (since \( (-1) \times (-5) = 5 \) and \( (-1) + (-5) = -6 \)).

Step 2: Rewrite the middle term and factor by grouping.

\( n^2 - n - 5n + 5 \)
\( = n(n - 1) - 5(n - 1) \)
\( = (n - 1)(n - 5) \)

Problem 12: \( x^2 + 3x - 18 \)

Step 1: Find two numbers that multiply to \( -18 \) and add to \( 3 \).

The numbers are \( 6 \) and \( -3 \) (since \( 6 \times (-3) = -18 \) and \( 6 + (-3) = 3 \)).

Step 2: Rewrite the middle term and factor by grouping.

\( x^2 + 6x - 3x - 18 \)
\( = x(x + 6) - 3(x + 6) \)
\( = (x + 6)(x - 3) \)

Problem 13: \( x^2 + 4x - 45 \)

Step 1: Find two numbers that multiply to \( -45 \) and add to \( 4 \).

The numbers are \( 9 \) and \( -5 \) (since \( 9 \times (-5) = -45 \) and \( 9 + (-5) = 4 \)).

Step 2: Rewrite the middle term and factor by grouping.

\( x^2 + 9x - 5x - 45 \)
\( = x(x + 9) - 5(x + 9) \)
\( = (x + 9)(x - 5) \)

Problem 14: \( k^2 + 12k + 32 \)

Step 1: Find two numbers that multiply to \( 32 \) and add to \( 12 \).

The numbers are \( 8 \) and \( 4 \) (since \( 8 \times 4 = 32 \) and \( 8 + 4 = 12 \)).

Step 2: Rewrite the middle term and factor by grouping.

\( k^2 + 8k + 4k + 32 \)
\( = k(k + 8) + 4(k + 8) \)
\( = (k + 8)(k + 4) \)

Problem 15: \( a^2 + 7a - 30 \)

Step 1: Find two numbers that multiply to \( -30 \) and add to \( 7 \).

The numbers are \( 10 \) and \( -3 \) (since \( 10 \times (-3) = -30 \) and \( 10 + (-3) = 7 \)).

Step 2: Rewrite the middle term and factor by grouping.

\( a^2 + 10a - 3a - 30 \)
\( = a(a + 10) - 3(a + 10) \)
\( = (a + 10)(a - 3) \)

Problem 16: \( x^2 - 3x - 10 \)

Step 1: Find two numbers that multiply to \( -10 \) and add to \( -3 \).

The numbers are \( -5 \) and \( 2 \) (since \( (-5) \times 2 = -10 \) and \( (-5) + 2 = -3 \)).

Step 2: Rewrite the middle term and factor by grouping.

\( x^2 - 5x + 2x - 10 \)
\( = x(x - 5) + 2(x - 5) \)
\( = (x - 5)(x + 2) \)

Problem 17: \( n^2 - 6n - 7 \)

Step 1: Find two numbers that multiply to \( -7 \) and add to \( -6 \).

The numbers are \( -7 \) and \( 1 \) (since \( (-7) \times 1 = -7 \) and \( (-7) + 1 = -6 \)).

Step 2: Rewrite the middle term and factor by grouping.

\( n^2 - 7n + n - 7 \)
\( = n(n - 7) + 1(n - 7) \)
\( = (n - 7)(n + 1) \)

Problem 18: \( p^2 - 18p + 80 \)

Step 1: Find two numbers that multiply to \( 80 \) and add to \( -18 \).

The numbers are \( -10 \) and \( -8 \) (since \( (-10) \times (-8) = 80 \) and \( (-10) + (-8) = -18 \)).

Step 2: Rewrite the middle term and factor by grouping.

\( p^2 - 10p - 8p + 80 \)
\( = p(p - 10) - 8(p - 10) \)
\( = (p - 10)(p - 8) \)

Problem 19: \( p^2 - 14p + 40 \)

Step 1: Find two numbers that multiply to \( 40 \) and add to \( -14 \).

The numbers are \( -10 \) and \( -4 \) (since \( (-10) \times (-4) = 40 \) and \( (-10) + (-4) = -14 \)).

Step 2: Rewrite the middle term and factor by grouping.

\( p^2 - 10p - 4p + 40 \)
\( = p(p - 10) - 4(p - 10) \)
\( = (p - 10)(p - 4) \)

Problem 20: \( a^2 - 14a + 45 \)

Step 1: Find two numbers that multiply to \( 45 \) and add to \( -14 \).

The numbers are \( -9 \) and \( -5 \) (since \( (-9) \times (-5) = 45 \) and \( (-9) + (-5) = -14 \)).

Step 2: Rewrite the middle term and factor by grouping.

\( a^2 - 9a - 5a + 45 \)
\( = a(a - 9) - 5(a - 9) \)
\( = (a - 9)(a - 5) \)

Final Answers:

1. \( \boldsymbol{(n - 1)(n - 5)} \)
2. \( \boldsymbol{(x + 6)(x - 3)} \)
3. \( \boldsymbol{(x + 9)(x - 5)} \)
4. \( \boldsymbol{(k + 8)(k + 4)} \)
5. \( \boldsymbol{(a + 10)(a - 3)} \)
6. \( \boldsymbol{(x - 5)(x + 2)} \)
7. \( \boldsymbol{(n - 7)(n + 1)} \)
8. \( \boldsymbol{(p - 10)(p - 8)} \)
9. \( \boldsymbol{(p - 10)(p - 4)} \)
10. \( \boldsymbol{(a - 9)(a - 5)} \)