Solving Rational and Radical Equations (Section 3.4) – Precalculus Study Notes

Section 3.4: Solving Rational Equations and Radical Equations

Objectives

• To solve rational equations.

• To solve radical equations.

Rational Equations

Definition and Key Concepts

A rational equation is an equation that contains one or more rational expressions (fractions with polynomials in the numerator and/or denominator).

• Solving rational equations typically involves eliminating denominators by multiplying both sides by the least common denominator (LCD) of all rational expressions.

• This process clears the equation of fractions, making it easier to solve for the variable.

Steps for Solving Rational Equations

1. Identify the LCD of all denominators in the equation.

2. Multiply both sides of the equation by the LCD to eliminate fractions.

3. Simplify and solve the resulting equation.

4. Check all solutions in the original equation to avoid extraneous solutions (values that make any denominator zero).

Example 1

Solve: $\dfrac{x-8}{3} + \dfrac{x-3}{2} = 0$

• LCD is 6.

• Multiply both sides by 6:

$6\left(\dfrac{x-8}{3} + \dfrac{x-3}{2}\right) = 6 \cdot 0$ $6 \cdot \dfrac{x-8}{3} + 6 \cdot \dfrac{x-3}{2} = 0$ $2(x-8) + 3(x-3) = 0$ $2x - 16 + 3x - 9 = 0$ $5x - 25 = 0$ $5x = 25$ $x = 5$

• Check $x = 5$ in the original equation:

$\dfrac{5-8}{3} + \dfrac{5-3}{2} = \dfrac{-3}{3} + \dfrac{2}{2} = -1 + 1 = 0$ Solution: $x = 5$

Example 2

Solve: $\dfrac{x^2}{x-3} = \dfrac{9}{x-3}$

• LCD is $x-3$.

• Multiply both sides by $x-3$:

$(x-3) \cdot \dfrac{x^2}{x-3} = (x-3) \cdot \dfrac{9}{x-3}$ $x^2 = 9$ $x = 3$ or $x = -3$

• Check $x = 3$:

• Denominator becomes zero ($x-3=0$), so $x=3$ is not a solution.

• Check $x = -3$:

• $\dfrac{(-3)^2}{-3-3} = \dfrac{9}{-6} = -1.5$ and $\dfrac{9}{-3-3} = \dfrac{9}{-6} = -1.5$; both sides equal, so $x = -3$ is a solution.

Solution: $x = -3$

Table: Checking Solutions for Rational Equations

x	Expression Value	Valid Solution?
5	0	Yes
3	Undefined (division by zero)	No
-3	Valid	Yes

Radical Equations

Definition and Key Concepts

A radical equation is an equation in which the variable appears in one or more radicands (expressions under a root, such as a square root).

• Example: $\sqrt{2x-5} - \sqrt{x-3} = 1$

• To solve, use the Principle of Powers: If $a = b$, then $a^n = b^n$ for any positive integer $n$.

Steps for Solving Radical Equations

1. Isolate the radical on one side of the equation.

2. Apply the Principle of Powers to eliminate the radical (usually by squaring both sides).

3. Solve the resulting equation.

4. If there are multiple radical terms, repeat the process as needed.

5. Check all solutions in the original equation to avoid extraneous solutions (values that do not satisfy the original equation).

Example 1

Solve: $\sqrt{3x+1} = 4$

• Square both sides:

$(\sqrt{3x+1})^2 = 4^2$ $3x+1 = 16$ $3x = 15$ $x = 5$

• Check $x = 5$:

• $\sqrt{3 \cdot 5 + 1} = \sqrt{16} = 4$

• True, so $x = 5$ is a solution.

Example 2

Solve: $5 + \sqrt{x+7} = x$

• Isolate the radical:

• $\sqrt{x+7} = x - 5$

• Square both sides:

$(\sqrt{x+7})^2 = (x-5)^2$ $x+7 = x^2 - 10x + 25$ $0 = x^2 - 11x + 18$ $0 = (x-9)(x-2)$ $x = 9$ or $x = 2$

• Check $x = 9$:

• $5 + \sqrt{9+7} = 5 + \sqrt{16} = 5 + 4 = 9$

• True, so $x = 9$ is a solution.

• Check $x = 2$:

• $5 + \sqrt{2+7} = 5 + \sqrt{9} = 5 + 3 = 8$

• Not equal to 2, so $x = 2$ is not a solution.

Table: Checking Solutions for Radical Equations

x	Expression Value	Valid Solution?
9	9	Yes
2	8	No

Important Notes

• Always check for extraneous solutions when solving rational and radical equations, as the process of squaring or multiplying by the LCD can introduce invalid solutions.

• Division by zero is undefined; any solution that makes a denominator zero must be excluded.

• For radical equations, ensure that the solution does not result in taking the square root of a negative number (unless working with complex numbers).

Summary Table: Rational vs. Radical Equations

Type of Equation	Key Feature	Solving Method	Check for Extraneous Solutions?
Rational Equation	Variable in denominator	Multiply by LCD	Yes
Radical Equation	Variable under a root	Isolate radical, apply powers	Yes