Solving Rational and Radical Equations

Definition The original domain of an equation in a real variable $x$ is the set of real numbers at which every expression in the equation, on both sides, is defined as a real number. For rational and radical equations this means that no denominator is $0$ and that the radicand of every even root (square root, fourth root and so on) is nonnegative. Odd roots such as $\sqrt[3]{x}$ add no restriction. A solution of the equation is a number in the original domain that makes the two sides equal.

Definition Two equations are equivalent on a set $D$ if they have exactly the same solutions in $D$. A step that turns an equation into one that is equivalent to it on the original domain $D$ is an equivalent transformation. The following steps are equivalent transformations.

1. Adding the same expression to both sides, or subtracting it, when the expression is defined at every point of $D$.
2. Multiplying or dividing both sides by the same nonzero number.
3. Multiplying both sides by an expression $g(x)$ that is defined and nonzero at every point of $D$.

Equivalence only speaks about numbers in $D$. The new equation may also be defined and true at numbers outside $D$, and those numbers are never solutions. Other common steps are not always equivalent. Multiplying both sides by an expression that is $0$ at some point of $D$, or squaring both sides, can add solutions. Dividing both sides by an expression that is $0$ at some point of $D$ can lose solutions.

Definition Factor every denominator completely. The least common denominator (LCD) of several rational expressions is the product of the least common multiple of the numerical factors and every distinct polynomial factor that appears, each raised to the highest power with which it occurs in any one denominator. Multiplying both sides of a rational equation by the LCD clears every fraction. The LCD is nonzero at every point of the original domain, so on that domain this is an equivalent transformation. The cleared equation, however, is defined for all real $x$, and it may have solutions where the LCD is $0$, which lie outside the original domain.

Definition For real numbers $a$ and $b$, if $a = b$ then $a^2 = b^2$, but $a^2 = b^2$ only says that $a = b$ or $a = -b$. So squaring both sides of $A(x) = B(x)$ keeps every solution and may add the solutions of $A(x) = -B(x)$. When $A(x) \ge 0$ and $B(x) \ge 0$ for every $x$ under consideration, squaring is an equivalent transformation. For an isolated square root this gives an exact rule,

The condition $A(x) \ge 0$ need not be added, because $A(x) = B(x)^2$ already forces it.

Definition An extraneous solution is a number that solves an equation reached later in the work but does not solve the original equation. Extraneous solutions have two sources. A step that is not an equivalent transformation, mainly squaring both sides, can add numbers inside the original domain. And a later equation may be defined at numbers outside the original domain and hold there. Clearing fractions does this, because the cleared equation is defined for every real $x$ and may hold where the LCD is $0$, and those are exactly the numbers a denominator excludes. Extraneous solutions are not arithmetic mistakes, so redoing the algebra carefully will not make them go away.

Definition Once the work includes squaring, clearing fractions or any other step that can bring in extraneous solutions, the numbers you find at the end are only candidates. A candidate $c$ is a solution exactly when $c$ lies in the original domain and substituting $c$ into the original equation, not into a later one, makes both sides equal. Substituting into the equation where a candidate appeared cannot expose it, because every candidate satisfies that equation. When the equation is $\sqrt{A(x)} = B(x)$ and a candidate $c$ satisfies $A(c) = B(c)^2$, testing $B(c) \ge 0$ is an exact check. This helps when $c$ is irrational, where a decimal substitution only gives approximate evidence.