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:
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\( 3x + 6 = 0 \)
Solving for \( x \): \[ 3x = -6 \implies x = -2 \]
-
\( x + 7 = 0 \)
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:
\[ x = -2 \quad \text{or} \quad x = -7 \]
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