Question

Solve the quadratic equation by grouping: 3x^2 + 11x + 6 = 0
What are solutions for x?

Answer 1

To solve the quadratic equation \(3x^2 + 11x + 6 = 0\) by grouping, we'll start by factoring it.

1. **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 \(9 + 2 = 11\).

2. **Rewrite the middle term using these numbers:**
\[
3x^2 + 9x + 2x + 6 = 0
\]

3. **Group the terms:**
\[
(3x^2 + 9x) + (2x + 6) = 0
\]

4. **Factor each group:**
\[
3x(x + 3) + 2(x + 3) = 0
\]

5. **Factor out the common binomial factor \((x + 3)\):**
\[
(3x + 2)(x + 3) = 0
\]

6. **Set each factor to zero:**
\[
3x + 2 = 0 \quad \text{or} \quad x + 3 = 0
\]

7. **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:
\[
x = -\frac{2}{3} \quad \text{and} \quad x = -3
\]