Question

Solve 4n^2+11n+6=0 by factoring. Enter your answers below
n=
or
n=

Answer 1

To solve the equation \( 4n^2 + 11n + 6 = 0 \) by factoring, we first need to find two numbers that multiply to \( 4 \times 6 = 24 \) and add to \( 11 \).

The numbers that satisfy these conditions are \( 3 \) and \( 8 \) (because \( 3 \times 8 = 24 \) and \( 3 + 8 = 11 \)).

Next, we can re-write the middle term of the quadratic:

\[
4n^2 + 3n + 8n + 6 = 0
\]

Now, we can group the terms:

\[
(4n^2 + 3n) + (8n + 6) = 0
\]

Factor out the common factors from each group:

\[
n(4n + 3) + 2(4n + 3) = 0
\]

Now, we can factor out \( (4n + 3) \):

\[
(4n + 3)(n + 2) = 0
\]

Now, we set each factor equal to zero:

1. \( 4n + 3 = 0 \)
\[
4n = -3 \implies n = -\frac{3}{4}
\]

2. \( n + 2 = 0 \)
\[
n = -2
\]

Thus, the solutions to the equation \( 4n^2 + 11n + 6 = 0 \) are:

\[
n = -\frac{3}{4} \quad \text{or} \quad n = -2
\]

So the final answers are:

\( n = -\frac{3}{4} \)

or

\( n = -2 \)