6 $8x^2 - 10x - 3 = 0$
7 $10x^2 + 5x - 18x - 9 = 0$
8 $6x^3 - 9x^2 - 60x = 0$
9 $2x^2 - 162 = 0$
10 $10x^3 + 45x^2 + 35x = 0$
$5x(2x^2 + 9 + 7) = 0$
$2x^3 + 2x + 7x + 7$
$2x(x + 1) + 7(x + 1)$
$5x(2x + 7)(x + 1)$
$5x = 0$ $2x + 7 = 0$ $x + 1 = 0$
extra scratch work

Question

6 $8x^2 - 10x - 3 = 0$
7 $10x^2 + 5x - 18x - 9 = 0$
8 $6x^3 - 9x^2 - 60x = 0$
9 $2x^2 - 162 = 0$
10 $10x^3 + 45x^2 + 35x = 0$
$5x(2x^2 + 9 + 7) = 0$
$2x^3 + 2x + 7x + 7$
$2x(x + 1) + 7(x + 1)$
$5x(2x + 7)(x + 1)$
$5x = 0$  $2x + 7 = 0$  $x + 1 = 0$
extra scratch work

Accepted Answer

Let's solve each equation one by one.

### Problem 6: Solve $8x^2 - 10x - 3 = 0$
# Explanation:
## Step 1: Factor the quadratic
We need to find two numbers that multiply to $8\times(-3)= -24$ and add up to $-10$. The numbers are $-12$ and $2$.
Rewrite the middle term: $8x^2 - 12x + 2x - 3 = 0$
Group the terms: $(8x^2 - 12x) + (2x - 3) = 0$
Factor out the GCF from each group: $4x(2x - 3) + 1(2x - 3) = 0$
Factor out $(2x - 3)$: $(4x + 1)(2x - 3) = 0$
## Step 2: Set each factor equal to zero
$4x + 1 = 0$ or $2x - 3 = 0$
For $4x + 1 = 0$: $4x = -1$ so $x = -\frac{1}{4}$
For $2x - 3 = 0$: $2x = 3$ so $x = \frac{3}{2}$
# Answer: $x = -\frac{1}{4}, \frac{3}{2}$

### Problem 7: Solve $10x^2 + 5x - 18x - 9 = 0$
# Explanation:
## Step 1: Simplify and factor
First, combine like terms: $10x^2 - 13x - 9 = 0$ (Wait, actually, let's factor by grouping as given: $10x^2 + 5x - 18x - 9 = 0$
Group: $(10x^2 + 5x) + (-18x - 9) = 0$
Factor: $5x(2x + 1) - 9(2x + 1) = 0$
Factor out $(2x + 1)$: $(5x - 9)(2x + 1) = 0$
## Step 2: Solve for $x$
$5x - 9 = 0$ or $2x + 1 = 0$
For $5x - 9 = 0$: $5x = 9$ so $x = \frac{9}{5}$
For $2x + 1 = 0$: $2x = -1$ so $x = -\frac{1}{2}$
# Answer: $x = \frac{9}{5}, -\frac{1}{2}$

### Problem 8: Solve $6x^3 - 9x^2 - 60x = 0$
# Explanation:
## Step 1: Factor out the GCF
The GCF of $6x^3$, $-9x^2$, and $-60x$ is $3x$.
Factor out $3x$: $3x(2x^2 - 3x - 20) = 0$
## Step 2: Factor the quadratic
Factor $2x^2 - 3x - 20$. We need two numbers that multiply to $2\times(-20)= -40$ and add up to $-3$. The numbers are $-8$ and $5$.
Rewrite the middle term: $2x^2 - 8x + 5x - 20 = 0$
Group: $(2x^2 - 8x) + (5x - 20) = 0$
Factor: $2x(x - 4) + 5(x - 4) = 0$
Factor out $(x - 4)$: $(2x + 5)(x - 4) = 0$
## Step 3: Set factors to zero
$3x = 0$ or $2x + 5 = 0$ or $x - 4 = 0$
For $3x = 0$: $x = 0$
For $2x + 5 = 0$: $2x = -5$ so $x = -\frac{5}{2}$
For $x - 4 = 0$: $x = 4$
# Answer: $x = 0, -\frac{5}{2}, 4$

### Problem 9: Solve $2x^2 - 162 = 0$
# Explanation:
## Step 1: Factor out the GCF
Factor out $2$: $2(x^2 - 81) = 0$
## Step 2: Factor the difference of squares
$x^2 - 81$ is a difference of squares: $x^2 - 9^2 = (x - 9)(x + 9)$
So the equation becomes $2(x - 9)(x + 9) = 0$
## Step 3: Set factors to zero
$x - 9 = 0$ or $x + 9 = 0$
For $x - 9 = 0$: $x = 9$
For $x + 9 = 0$: $x = -9$
# Answer: $x = 9, -9$

### Problem 10: Solve $10x^3 + 45x^2 + 35x = 0$ (Correcting the initial factoring mistake)
# Explanation:
## Step 1: Factor out the GCF
The GCF of $10x^3$, $45x^2$, and $35x$ is $5x$.
Factor out $5x$: $5x(2x^2 + 9x + 7) = 0$
## Step 2: Factor the quadratic
Factor $2x^2 + 9x + 7$. We need two numbers that multiply to $2\times7 = 14$ and add up to $9$. The numbers are $2$ and $7$.
Rewrite the middle term: $2x^2 + 2x + 7x + 7 = 0$
Group: $(2x^2 + 2x) + (7x + 7) = 0$
Factor: $2x(x + 1) + 7(x + 1) = 0$
Factor out $(x + 1)$: $(2x + 7)(x + 1) = 0$
## Step 3: Set factors to zero
$5x = 0$ or $2x + 7 = 0$ or $x + 1 = 0$
For $5x = 0$: $x = 0$
For $2x + 7 = 0$: $2x = -7$ so $x = -\frac{7}{2}$
For $x + 1 = 0$: $x = -1$
# Answer: $x = 0, -\frac{7}{2}, -1$

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QUESTION IMAGE

Question

Explanation:

Problem 6: Solve \(8x^2 - 10x - 3 = 0\)

Step 1: Factor the quadratic

Step 2: Set each factor equal to zero

Step 1: Simplify and factor

Step 2: Solve for \(x\)

Step 1: Factor out the GCF

Step 2: Factor the quadratic

Step 3: Set factors to zero

Answer:

Problem 7: Solve \(10x^2 + 5x - 18x - 9 = 0\)