Question

To show how to solve the equation, create two functions and find their intersection points. What two functions can be used to solve the following equation, and what is their solution set?

x2+4x−1=2x+2
Responses:

f(x)=x^2+4x−1, g(x)=2x+2, {−3,−4,1,4}

f(x)=x^2+4x−1, g(x)=2x+2, {−3,1}

f(x)=x^2+4x, g(x)=2x, {−3,1}

f(x)=x^2+4x−1, g(x)=2x+2, {−1,3}

Answer 1

To solve the equation \( x^2 + 4x - 1 = 2x + 2 \), we can rearrange the equation into a standard form.

1. **Rearranging the Equation**:

Start by moving all terms to one side:
\[
x^2 + 4x - 1 - 2x - 2 = 0
\]
This simplifies to:
\[
x^2 + 2x - 3 = 0
\]

2. **Identifying the Functions**:

We can define two functions:
- \( f(x) = x^2 + 4x - 1 \)
- \( g(x) = 2x + 2 \)

These functions represent the left and right sides of our original equation.

3. **Finding Intersection Points**:

The intersection points of these functions correspond to the solutions of the equation \( f(x) = g(x) \). Setting the functions equal gives us:
\[
x^2 + 4x - 1 = 2x + 2
\]
Rearranging gives:
\[
x^2 + 2x - 3 = 0
\]

4. **Factoring the Quadratic**:
Next, we factor the quadratic:
\[
(x + 3)(x - 1) = 0
\]

5. **Finding the Roots**:

From the factored form, the solutions are:
\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]
\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]

Thus, the solution set is \(\{-3, 1\}\).

Based on this analysis, the correct response is:
**f(x)=x^2+4x−1, g(x)=2x+2, {−3,1}**.