ANSWER
The correct answer is
\(y = {x}^{2}\)
and
\(y = 2x + 3\)
EXPLANATION
The given equation is
\({x}^{2} = 2x + 3\)
If we consider this equation as a single equation, then we can use methods such as factorization, completing the squares, or the quadratic formula to solve it.
For instance using the method of factorization we proceed as follows,
\({x}^{2} - 2x - 3 = 0\)
\({x}^{2} - 3x + x - 3 = 0\)
\(x(x - 3) + 1(x - 3) = 0\)
\((x - 3)(x + 1) = 0\)
\(x = - 1 \: or \: x = 3\)
Meaning when we graph the function
\(y = {x}^{2} - 2x - 3\)
, it will intersect the x-axis at
\(x = - 1 \: and \: x = 3\)
However, if we consider the equation to be two different systems combined, that is the equation of the straight line
\(y = 2x + 3\)
and the equation of the basic quadratic function
\(y = {x}^{2}\)
Then the solution to
\({x}^{2} = 2x + 3\)
is the intersection of the graph of the straight line and the graph of the quadratic curve.
That is the x-coordinates of the points of intersection are the solutions.
See graph.
