To solve the quadratic equation \(3x^2 + 11x + 6 = 0\) by grouping, we'll start by factoring it.
-
Find two numbers that multiply to \(3 \times 6 = 18\) (the product of the coefficient of \(x^2\) and the constant term) and add to \(11\) (the coefficient of \(x\)).
- The numbers that satisfy these conditions are \(9\) and \(2\) since \(9 \times 2 = 18\) and \(9 + 2 = 11\).
-
Rewrite the middle term using these numbers: \[ 3x^2 + 9x + 2x + 6 = 0 \]
-
Group the terms: \[ (3x^2 + 9x) + (2x + 6) = 0 \]
-
Factor each group: \[ 3x(x + 3) + 2(x + 3) = 0 \]
-
Factor out the common binomial factor \((x + 3)\): \[ (3x + 2)(x + 3) = 0 \]
-
Set each factor to zero: \[ 3x + 2 = 0 \quad \text{or} \quad x + 3 = 0 \]
-
Solve for \(x\):
- 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: \[ x = -\frac{2}{3} \quad \text{and} \quad x = -3 \]