Given the graph of the equation $y = 2x^3 - x^2 + 3$:
We want to solve the equation $2x^3 - 7 = 0$ through graphical methods.
In the solution they do the following:
$2x^3 - x^2 + 3 = -x^2 + 10$
They then graph the quadratic $-x^2 + 10$ and find the intersection between the two graphs. Yet how does the above method work? What is the intuition behind solving these equations graphically?
Eg. When we are given the graph $y = \frac12x^2 + 1$ and we want to solve $\frac12x^2 - 2 = 0$. Why can we draw a line at $y = 3$ and find the intersection? I can't seem to find the intuition.
$$2x^3 - x^2 + 3 = (2x^3 - 7) + [7 + (-x^2 + 3)]. \tag1 $$
In (1) above, the presumption that $2x^3 - 7 = 0$ implies that the first term on the RHS is $(0)$.
As a consequence of this implication, you can deduce that
$$2x^3 - x^2 + 3 = (0) + [7 + (-x^2 + 3)]. \tag2 $$
(2) above implies that any point $(x,y)$ that satisfies the equation will be an intersection point of two different graphs:
The graph of $f(x) = 2x^3 - x^2 + 3$.
The graph of $g(x) = -x^2 + 10.$
That is, any value of $x$ that satisfies (2) above will be such that $f(x) = g(x).$
Since $f(x)$ employs a 3rd degree polynomial, and since 3rd degree polynomial equations are often harder to solve than quadratic equations, this graphical approach makes it easier for you to visualize approximately where the pertinent value(s) of $(x)$ is/are located.
It turns out that with this particular problem, the equation $f(x) = g(x)$ will be true, if and only if $2x^3 = 7.$
Here, the only Real number $x$ that satisfies $f(x) = g(x)$ is easily seen to be $\displaystyle \left[\frac{7}{2}\right]^{(1/3)}.$
So, the problem composer intentionally provided you with an example, where straight algebra could have been used instead. The problem composer's thinking is that this would sanity-check that the solution given as the approximate location of the intersection of the graphs is reasonably accurate.
In other words, by carefully crafting this particular problem, the problem composer hopes that the student will gain confidence in the concept that graphical methods can be used to approximate the Real root(s) of polynomial equations of degree, higher than $(2)$.