Question

Solve the quadratic equation by grouping:

3x2+11x+6=0


What are the solutions for x?

(1 point)
Responses

x=3 and x=23
x=3 and x=2 thirds

x=−32
and x=-3
x=negative 3 halves and x=-3

x=−23
and x=-3
x=negative 2 thirds and x=-3

x=-2 and x=-3

Answer 1

To solve the quadratic equation \( 3x^2 + 11x + 6 = 0 \) by grouping, we can follow these steps:

1. **Multiply the coefficient of \( x^2 \) (which is 3) by the constant term (which is 6)**:
\[
3 \times 6 = 18
\]

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

3. **Rewrite the middle term (11x) using 9x and 2x**:
\[
3x^2 + 9x + 2x + 6 = 0
\]

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

5. **Factor out the common factors in each group**:
\[
3x(x + 3) + 2(x + 3) = 0
\]

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

7. **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 \( x + 3 = 0 \):
\[
x = -3
\]

8. **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**