The image presents a set of quadratic equations that need to be solved by factoring. I will solve problems 4, 5, 6, 7, and 8. Problem 3 is already solved.

1. Problem Description

The image presents a set of quadratic equations that need to be solved by factoring. I will solve problems 4, 5, 6, 7, and

8. Problem 3 is already solved.

2. Solution Steps

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

Step 1: Divide the entire equation by 7 to simplify it.

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

Step 2: Factor the quadratic expression.

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

Step 3: Set each factor equal to zero and solve for $x$.

$x+7 = 0$ or $x-1 = 0$

$x = -7$ or $x = 1$

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

Step 1: Add 3 to both sides of the equation to set it equal to zero.

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

Step 2: Divide the entire equation by 4 to simplify it.

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

Step 3: Factor the quadratic expression.

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

Step 4: Set each factor equal to zero and solve for $x$.

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

$2x = 1$ or $x = 4$

$x = \frac{1}{2}$ or $x = 4$

Problem 6: $6a^2 - 3a - 48 = -3$

Step 1: Add 3 to both sides of the equation to set it equal to zero.

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

Step 2: Divide the entire equation by 3 to simplify it.

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

Step 3: Factor the quadratic expression.

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

Step 4: Set each factor equal to zero and solve for $a$.

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

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

$a = -\frac{5}{2}$ or $a = 3$

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

Step 1: Subtract 8 from both sides of the equation to set it equal to zero.

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

Step 2: Divide the entire equation by 6 to simplify it.

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

Step 3: Factor the quadratic expression.

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

Step 4: Set each factor equal to zero and solve for $x$.

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

$7x = 3$ or $x = 3$

$x = \frac{3}{7}$ or $x = 3$

Problem 8: $10n^2 - 10n = -5n$

Step 1: Add $5n$ to both sides of the equation to set it equal to zero.

$10n^2 - 5n = 0$

Step 2: Factor out the common term $5n$.

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

Step 3: Set each factor equal to zero and solve for $n$.

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

$n = 0$ or $2n = 1$

$n = 0$ or $n = \frac{1}{2}$

3. Final Answer

Problem 4: $x = -7$ or $x = 1$

Problem 5: $x = \frac{1}{2}$ or $x = 4$

Problem 6: $a = -\frac{5}{2}$ or $a = 3$

Problem 7: $x = \frac{3}{7}$ or $x = 3$

Problem 8: $n = 0$ or $n = \frac{1}{2}$