Question

which statement correctly interprets the solution to the equation x^2 + 5x + 6 = -3x^2 - 2x

An input of -2 will yield an outpost of 0 on both sides of the equation
An input of -3 will yield and outpost of 0 on both sides of the equation
An input of 0 will yield an input of 0 on both sides of the equation
There are no input values that would yield the same output Callie on both sides of the equation

Answer 1

To interpret the solution to the equation \( x^2 + 5x + 6 = -3x^2 - 2x \), we first need to rearrange the equation into standard form and then solve for \( x \).

Starting with the given equation:

\[
x^2 + 5x + 6 + 3x^2 + 2x = 0
\]

Combine like terms:

\[
4x^2 + 7x + 6 = 0
\]

Next, we can factor the quadratic equation \( 4x^2 + 7x + 6 = 0 \).

To factor, we look for two numbers that multiply to \( 4 \times 6 = 24 \) and add to \( 7 \). These numbers are \( 3 \) and \( 4 \). We can express the quadratic as follows:

\[
4x^2 + 3x + 4x + 6 = 0
\]

Grouping:

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

Factoring by grouping gives:

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

Factoring out the common term \( (4x + 3) \):

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

Setting each factor to zero, we find the solutions:

1. \( 4x + 3 = 0 \) implies \( x = -\frac{3}{4} \)
2. \( x + 2 = 0 \) implies \( x = -2 \)

From the context of the choices provided:

- An input of -2 will yield an output of 0 on both sides of the equation.
- An input of -3 will yield an output of 0 on both sides of the equation.
- An input of 0 will yield an output of 0 on both sides of the equation.
- There are no input values that would yield the same output on both sides of the equation.

The correct answer is: **An input of -2 will yield an output of 0 on both sides of the equation.**

When \( x = -2 \):

Left side: \( (-2)^2 + 5(-2) + 6 = 4 - 10 + 6 = 0 \)
Right side: \( -3(-2)^2 - 2(-2) = -3(4) + 4 = -12 + 4 = -8 \)

Therefore, only the left side equals 0, confirming that the input value of -2 does yield an output of 0 on the left side, but not on both as stated in your option. Hence, the interpretation remains accurate that \( x = -2 \) makes the left side equal to zero.