Which system of equations can be graphed to find the solution(s) to [tex][tex]$x^2 = 2x + 3$[/tex][/tex]?

A. [tex]\left\{\begin{array}{l}y = x^2 + 2x + 3 \\ y = 2x + 3\end{array}\right.[/tex]
B. [tex]\left\{\begin{array}{l}y = x^2 - 3 \\ y = 2x + 3\end{array}\right.[/tex]
C. [tex]\left\{\begin{array}{l}y = x^2 - 2x - 3 \\ y = 2x + 3\end{array}\right.[/tex]
D. [tex]\left\{\begin{array}{l}y = x^2 \\ y = 2x + 3\end{array}\right.[/tex]



Answer :

To solve the equation [tex]\(x^2 = 2x + 3\)[/tex] by finding the system of equations that can be graphed, we want to express both sides of the equation as separate functions of [tex]\(x\)[/tex]. Then, we'll see which system of equations involves these functions.

Starting from the given equation:
[tex]\[ x^2 = 2x + 3 \][/tex]

We can rearrange the terms to form:
[tex]\[ x^2 - 2x - 3 = 0 \][/tex]

Now we need to find a system of equations where one equation represents [tex]\(y = x^2\)[/tex] and the other represents [tex]\(y = 2x + 3\)[/tex].

Let's look at the options provided:

1. [tex]\(\left\{\begin{array}{l} y = x^2 + 2x + 3 \\ y = 2x + 3 \end{array}\right.\)[/tex]

2. [tex]\(\left\{\begin{array}{l} y = x^2 - 3 \\ y = 2x + 3 \end{array}\right.\)[/tex]

3. [tex]\(\left\{\begin{array}{l} y = x^2 - 2x - 3 \\ y = 2x + 3 \end{array}\right.\)[/tex]

4. [tex]\(\left\{\begin{array}{l} y = x^2 \\ y = 2x + 3 \end{array}\right.\)[/tex]

We can see that the correct system should include the equation [tex]\(y = x^2\)[/tex] on one side and [tex]\(y = 2x + 3\)[/tex] on the other side.

Upon comparing and analysis of each options:
1. [tex]\(\left\{\begin{array}{l} y = x^2 + 2x + 3 \\ y = 2x + 3 \end{array}\right.\)[/tex] is incorrect because of the left part [tex]\(x^2 + 2x + 3 \neq x^2\)[/tex]
2. [tex]\(\left\{\begin{array}{l} y = x^2 - 3 \\ y = 2x + 3 \end{array}\right.\)[/tex] doesn't match the rearranged [tex]\(x^2 - 2x - 3 \equiv 0\)[/tex]
3. [tex]\(\left\{\begin{array}{l} y = x^2 - 2x - 3 \\ y = 2x + 3 \end{array}\right.\)[/tex] rearranged terms PERFECTLY combines both full equations.
4. [tex]\(\left\{\begin{array}{l} y = x^2 \\ y = 2x + 3 \end{array}\right.\)[/tex] is incomplete, as it misses the component of matching BOTH equations.

Thus, the correct system of equations to graph in order to find the solutions to [tex]\(x^2 = 2x + 3\)[/tex] is:
[tex]\[ \left\{\begin{array}{l} y = x^2 - 2x - 3 \\ y = 2x + 3 \end{array}\right. \][/tex]