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=-2 and x=-3
x=-2 and x=-3

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

x=−32
and x=-3

Answer 1

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:

1. \(3x + 2 = 0\)
\[
3x = -2 \implies x = -\frac{2}{3}
\]

2. \(x + 3 = 0\)
\[
x = -3
\]

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 \(x = -3\)