To solve the quadratic equation \( 3x^2 + 11x + 6 = 0 \) by grouping, we will begin by splitting the middle term based on the product of the coefficient of \( x^2 \) and the constant term.
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Multiply the coefficient of \( x^2 \) (which is 3) and the constant term (which is 6): \[ 3 \cdot 6 = 18 \]
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We need to find two numbers that multiply to 18 and add up to 11 (the coefficient of \( x \)). The two numbers that satisfy this are 9 and 2.
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Rewrite the middle term (11x) using 9x and 2x: \[ 3x^2 + 9x + 2x + 6 = 0 \]
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Now group the terms: \[ (3x^2 + 9x) + (2x + 6) = 0 \]
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Factor out the common factors from each group: \[ 3x(x + 3) + 2(x + 3) = 0 \]
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Now we can factor by grouping: \[ (3x + 2)(x + 3) = 0 \]
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Setting each factor equal to zero gives us: \[ 3x + 2 = 0 \quad \text{or} \quad x + 3 = 0 \]
From \( 3x + 2 = 0 \): \[ 3x = -2 \quad \Rightarrow \quad x = -\frac{2}{3} \]
From \( x + 3 = 0 \): \[ x = -3 \]
Thus, the solutions for \( x \) are: \[ x = -\frac{2}{3} \quad \text{and} \quad x = -3 \]
Therefore, the correct answer is B) x = -2/3 and x = -3.
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