What are the solutions to the system of equations?

[tex]\[
\begin{array}{l}
y = x^2 + 5x + 6 \\
y = 3x + 6
\end{array}
\][/tex]

A. [tex]\((0,6)\)[/tex] and [tex]\((-2,0)\)[/tex]

B. [tex]\((0,-3)\)[/tex] and [tex]\((6,0)\)[/tex]

C. [tex]\((-3,0)\)[/tex] and [tex]\((0,6)\)[/tex]

D. [tex]\((-3,0)\)[/tex] and [tex]\((-2,0)\)[/tex]



Answer :

To find the solutions to the system of equations:
[tex]\[ \begin{array}{l} y = x^2 + 5x + 6 \\ y = 3x + 6 \end{array} \][/tex]
we need to solve these equations simultaneously.

### Step-by-Step Solution:

1. Set the equations equal to each other:

[tex]\[ x^2 + 5x + 6 = 3x + 6 \][/tex]

2. Simplify the equation:

Subtract [tex]\(3x + 6\)[/tex] from both sides:

[tex]\[ x^2 + 5x + 6 - (3x + 6) = 0 \implies x^2 + 2x = 0 \][/tex]

3. Factor the quadratic equation:

[tex]\[ x(x + 2) = 0 \][/tex]

4. Solve for [tex]\(x\)[/tex]:

The solutions to the factored equation are:

[tex]\[ x = 0 \quad \text{or} \quad x = -2 \][/tex]

5. Determine the corresponding [tex]\(y\)[/tex]-values for each [tex]\(x\)[/tex]:

- When [tex]\(x = 0\)[/tex]:
[tex]\[ y = 3x + 6 = 3(0) + 6 = 6 \][/tex]
Thus, one solution is [tex]\((0, 6)\)[/tex].

- When [tex]\(x = -2\)[/tex]:
[tex]\[ y = 3x + 6 = 3(-2) + 6 = -6 + 6 = 0 \][/tex]
Thus, the other solution is [tex]\((-2, 0)\)[/tex].

### Solutions:

The solutions to the system of equations are [tex]\((0, 6)\)[/tex] and [tex]\((-2, 0)\)[/tex]. Therefore, the correct answer is:

[tex]\[ (0,6) \text{ and } (-2,0) \][/tex]