To solve the quadratic equation \( 3x^2 + 11x + 6 = 0 \) by grouping, we can follow these steps:
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Multiply the coefficient of \( x^2 \) (which is 3) by the constant term (which is 6): \[ 3 \times 6 = 18 \]
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We need to find two numbers that multiply to 18 and add to 11: The numbers are 9 and 2, since \( 9 \times 2 = 18 \) and \( 9 + 2 = 11 \).
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Rewrite the middle term (11x) using 9x and 2x: \[ 3x^2 + 9x + 2x + 6 = 0 \]
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Group the terms: \[ (3x^2 + 9x) + (2x + 6) = 0 \]
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Factor out the common factors in each group: \[ 3x(x + 3) + 2(x + 3) = 0 \]
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Factor out the common binomial factor: \[ (3x + 2)(x + 3) = 0 \]
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Set each factor equal to zero: \[ 3x + 2 = 0 \quad \text{or} \quad x + 3 = 0 \]
From \( 3x + 2 = 0 \): \[ 3x = -2 \ x = -\frac{2}{3} \]
From \( x + 3 = 0 \): \[ x = -3 \]
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The solutions for \( x \) are: \[ x = -\frac{2}{3} \quad \text{and} \quad x = -3 \]
Thus, the answer is:
- x = negative 2 thirds and x = -3
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