To solve the quadratic equation \(3x^2 + 11x + 6 = 0\) by grouping, we can start by looking for two numbers that multiply to give \(3 \cdot 6 = 18\) (the product of the coefficient of \(x^2\) and the constant term) and add up to give \(11\) (the coefficient of \(x\)).
The two numbers that satisfy these conditions are \(2\) and \(9\) because:
- \(2 \times 9 = 18\)
- \(2 + 9 = 11\)
Next, we can split the middle term (\(11x\)) using these two numbers:
\[ 3x^2 + 2x + 9x + 6 = 0 \]
Now we can group the terms:
\[ (3x^2 + 2x) + (9x + 6) = 0 \]
Factoring each group gives:
\[ x(3x + 2) + 3(3x + 2) = 0 \]
Now, we can factor out the common term \((3x + 2)\):
\[ (3x + 2)(x + 3) = 0 \]
Now we can set each factor to zero:
-
\(3x + 2 = 0\)
- \(3x = -2\)
- \(x = -\frac{2}{3}\)
-
\(x + 3 = 0\)
- \(x = -3\)
Thus, the solutions for \(x\) are:
\[ x = -\frac{2}{3} \quad \text{and} \quad x = -3 \]