To solve the quadratic equation \(3x^2 + 11x + 6 = 0\) by grouping, we first look for two numbers that multiply to \(3 \times 6 = 18\) and add to \(11\). The numbers \(9\) and \(2\) fit this requirement, as \(9 + 2 = 11\) and \(9 \times 2 = 18\).
Next, we can rewrite the middle term of the quadratic equation:
\[ 3x^2 + 9x + 2x + 6 = 0 \]
Now, we group the terms:
\[ (3x^2 + 9x) + (2x + 6) = 0 \]
Factoring each group, we have:
\[ 3x(x + 3) + 2(x + 3) = 0 \]
Now, we can factor out the common factor \((x + 3)\):
\[ (3x + 2)(x + 3) = 0 \]
Now, we can set each factor equal to zero:
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\(3x + 2 = 0\) \[ 3x = -2 \implies x = -\frac{2}{3} \]
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\(x + 3 = 0\) \[ x = -3 \]
Thus, the solutions to the equation \(3x^2 + 11x + 6 = 0\) are:
\[ x = -\frac{2}{3} \quad \text{and} \quad x = -3 \]
So the correct responses are:
- \(x = -\frac{2}{3}\) and \(x = -3\)
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