Question

Use factoring to solve the equation
3xsquaredplus25xequalsminus42
.
.
.
Question content area right
Part 1
The solution is xequals
  
enter your response here.

Answer 1

To solve the equation \( 3x^2 + 25x = -42 \) using factoring, we first reorganize it to set it to zero:

\[
3x^2 + 25x + 42 = 0
\]

Now, we can factor this quadratic equation. We need to find two numbers that multiply to \( 3 \times 42 = 126 \) and add up to \( 25 \).

The pairs of factors of \( 126 \) that we can consider are:
- \( 1 \times 126 \)
- \( 2 \times 63 \)
- \( 3 \times 42 \)
- \( 6 \times 21 \)
- \( 7 \times 18 \)
- \( 9 \times 14 \)

The pair that adds up to \( 25 \) is \( 21 \) and \( 6 \).

Now, we can rewrite the middle term of the quadratic using these two numbers:

\[
3x^2 + 21x + 6x + 42 = 0
\]

Next, we group the terms:

\[
(3x^2 + 21x) + (6x + 42) = 0
\]

Now we factor each group:

\[
3x(x + 7) + 6(x + 7) = 0
\]

Now we factor out the common factor:

\[
(3x + 6)(x + 7) = 0
\]

Now we set each factor to zero:

1. \( 3x + 6 = 0 \)

Solving for \( x \):
\[
3x = -6 \implies x = -2
\]

2. \( x + 7 = 0 \)

Solving for \( x \):
\[
x = -7
\]

Thus, the solutions to the equation are:

\[
x = -2 \quad \text{and} \quad x = -7
\]

So, the final answers are:

\[
x = -2 \quad \text{or} \quad x = -7
\]