College Algebra Chapter 1.6: Other Types of Equations - Study Notes

Equations and Inequalities

Other Types of Equations

This section covers advanced equation types commonly encountered in college algebra, including polynomial, radical, rational exponent, quadratic-form, and absolute value equations. Mastery of these techniques is essential for solving a wide variety of algebraic problems and applications.

Polynomial Equations

Polynomial equations are formed by setting two polynomials equal to each other. The equation is in standard form if one side is zero and the other side is a polynomial arranged in descending powers of the variable. The degree of the equation is the highest degree of any term present.

• Key Point: To solve polynomial equations, move all terms to one side to set the equation equal to zero.

• Key Point: Factor the polynomial, set each factor equal to zero, and solve for the variable.

• Example: Solve $x^3 - 4x^2 + x + 6 = 0$ by factoring and finding roots.

Radical Equations

Radical equations contain variables within a square root, cube root, or higher root. To solve, isolate the radical and raise both sides to the appropriate power to eliminate the root.

• Key Point: Isolate the radical on one side before raising both sides to the nth power.

• Key Point: Always check proposed solutions in the original equation, as extraneous solutions may arise.

• Example: Solve $\sqrt{x+1} = x-1$ by isolating the radical and squaring both sides.

Equations with Rational Exponents

Rational exponents represent radicals, e.g., $x^{1/n} = \sqrt[n]{x}$. To solve equations with rational exponents, isolate the expression and raise both sides to the reciprocal power.

• Key Point: Isolate the term with the rational exponent.

• Key Point: Raise both sides to the reciprocal of the exponent to eliminate it.

• Example: Solve $x^{2/3} = 8$ by raising both sides to the $3/2$ power.

Equations Quadratic in Form

Some equations can be rewritten as quadratic equations using substitution. This is useful when the equation is not explicitly quadratic but can be transformed into one.

• Key Point: Identify a substitution that transforms the equation into quadratic form.

• Key Point: Solve the quadratic equation, then substitute back to find the original variable.

• Example: Solve $x^4 - 5x^2 + 6 = 0$ by letting $u = x^2$.

Equations Involving Absolute Value

The absolute value of $x$ is the distance from zero on the number line. To solve absolute value equations, rewrite them without absolute value bars, considering both positive and negative cases.

• Key Point: If $|u| = c$, then $u = c$ or $u = -c$ (for $c > 0$).

• Key Point: Solve both resulting equations and check for extraneous solutions.

• Example: Solve $|x-3| = 5$ by considering $x-3 = 5$ and $x-3 = -5$.

Applications of Equations

Equations are used to model real-world situations, such as predicting weekly television viewing time based on annual income. Solving such equations involves substituting known values and solving for the unknown variable.

• Key Point: Translate the problem into an equation and solve for the desired variable.

• Example: Given $H = -0.0002I^2 + 0.065I + 27.4$, find $I$ when $H = 33.1$.