Solving Equations That Use Other Algebraic Functions

Rational Functions in Equations
An equation involving only rational functions can always be expressed in the form: $R(x) = \frac {P(x)}{Q(x)}= 0$.
A solution, $x=a$, must also be a solution for the polynomial equation $P(x) = 0$.

Thus solving rational function equations reduces to solving polynomial equations.
The Factor Theorem gives the result that connects solving polynomial equation with the factors  of a polynomial functions.

Beginning algebra courses spend a considerable amount of time solving quadratic equations without any reference to functions.

Solving equations with linear fractional expressions is also a subject of beginning and intermediate algebra courses.

Other Algebraic Functions in Equations
Equations involving roots can be more challenging the rational functions. They are more likely to be considered in courses at least at the intermediate algebra level.

Here are some examples that illustrate the visualization of equation solutions with mapping diagrams and graphs.

Example OAF.SAE.1 : Suppose $\frac {x^2 - 4}{x^2+1} = 0$. Find $x$.

Example OAF.SAE.2 : Suppose $\sqrt {x + 5} = 2x - 5$. Find $x$.
You can use this next dynamic example to visualize solving algebraic equations like those in Examples QEQ.1 and QEQ.2 with a mapping diagram and graph.

Example OAF.DSAE.0 Dynamic Views for solving an equation $f(x) = g(x)$ and $f(x) = \frac {P(x)}{Q(x)}$ on Graphs and Mapping Diagrams