We are asked to solve the following quadratic equations by factoring: 3) $6a^2 - 18a - 108 = 0$ 4) $7x^2 + 42x - 49 = 0$ 5) $8x^2 - 36x + 13 = -3$ 6) $6a^2 - 3a - 48 = -3$ 7) $42x^2 - 144x + 62 = 8$ 8) $10n^2 - 10n = -5n$

1. Problem Description

We are asked to solve the following quadratic equations by factoring:

3) $6a^2 - 18a - 108 = 0$

4) $7x^2 + 42x - 49 = 0$

5) $8x^2 - 36x + 13 = -3$

6) $6a^2 - 3a - 48 = -3$

7) $42x^2 - 144x + 62 = 8$

8) $10n^2 - 10n = -5n$

2. Solution Steps

3) $6a^2 - 18a - 108 = 0$

First, divide by 6:

$a^2 - 3a - 18 = 0$

Now, factor the quadratic:

$(a - 6)(a + 3) = 0$

So, $a = 6$ or $a = -3$.

4) $7x^2 + 42x - 49 = 0$

Divide by 7:

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

Factor the quadratic:

$(x + 7)(x - 1) = 0$

So, $x = -7$ or $x = 1$.

5) $8x^2 - 36x + 13 = -3$

Add 3 to both sides:

$8x^2 - 36x + 16 = 0$

Divide by 4:

$2x^2 - 9x + 4 = 0$

Factor the quadratic:

$(2x - 1)(x - 4) = 0$

So, $2x - 1 = 0$ or $x - 4 = 0$

$x = 1/2$ or $x = 4$.

6) $6a^2 - 3a - 48 = -3$

Add 3 to both sides:

$6a^2 - 3a - 45 = 0$

Divide by 3:

$2a^2 - a - 15 = 0$

Factor the quadratic:

$(2a + 5)(a - 3) = 0$

So, $2a + 5 = 0$ or $a - 3 = 0$

$a = -5/2$ or $a = 3$.

7) $42x^2 - 144x + 62 = 8$

Subtract 8 from both sides:

$42x^2 - 144x + 54 = 0$

Divide by 6:

$7x^2 - 24x + 9 = 0$

Factor the quadratic:

$(7x - 3)(x - 3) = 0$

So, $7x - 3 = 0$ or $x - 3 = 0$

$x = 3/7$ or $x = 3$.

8) $10n^2 - 10n = -5n$

Add $5n$ to both sides:

$10n^2 - 5n = 0$

Factor out $5n$:

$5n(2n - 1) = 0$

So, $5n = 0$ or $2n - 1 = 0$

$n = 0$ or $n = 1/2$.

3. Final Answer

3) $a = 6, -3$

4) $x = -7, 1$

5) $x = \frac{1}{2}, 4$

6) $a = -\frac{5}{2}, 3$

7) $x = \frac{3}{7}, 3$

8) $n = 0, \frac{1}{2}$