Section 1.6 Other Types of Equations

Polynomial Equations

Example Solve by Factoring:

Example Solve by Factoring:

Radical Equations

A radical equation is an equation in which the variable occurs in a square root, cube root, or any higher root. We solve the equation by squaring both sides.

This new equation has two solutions, -4 and 4 This new equation has two solutions, -4 and 4. By contrast, only 4 is a solution of the original equation, x=4. For this reason, when raising both sides of an equation to an even power, check proposed solutions in the original equation. Extra solutions may be introduced when you raise both sides of a radical equation to an even power. Such solutions, which are not solutions of the given equation are called extraneous solutions or extraneous roots.

Example Solve and check your answers:

Solving an Equation That Has Two Radicals Isolate a radical on one side. Square both sides. Repeat Step 1: Isolate the remaining radical on one side. Repeat Step2: Square both sides. Solve the resulting equation Check the proposed solutions in the original equations.

Example Solve:

Equations with Rational Exponents

Example Solve:

Equations That Are Quadratic in Form

Some equations that are not quadratic can be written as quadratic equations using an appropriate substitution. Here are some examples: An equation that is quadratic in form is one that can be expressed as a quadratic equation using an appropriate substitution.

Example Simplify:

Example Simplify:

Equations Involving Absolute Value

Example Solve:

Absolute Value Graphs The graph may intersect the x axis at one point, no points or two points. Thus the equations could have one, or two solutions or no solutions.

Solve, and check your solutions: (b) (c) (d)

Solve: (a) (b) (c) (d)