Key Concepts

• Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve a radical equation, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1.
• Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping.
• We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index.
• To solve absolute value equations, we need to write two equations, one for the positive value and one for the negative value.
• Equations in quadratic form are easy to spot, as the exponent on the first term is double the exponent on the second term and the third term is a constant. We may also see a binomial in place of the single variable. We use substitution to solve.
• Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form.

Glossary

absolute value equation

an equation in which the variable appears in absolute value bars, typically with two solutions, one accounting for the positive expression and one for the negative expression

equations in quadratic form

equations with a power other than 2 but with a middle term with an exponent that is one-half the exponent of the leading term

extraneous solutions

any solutions obtained that are not valid in the original equation

polynomial equation

an equation containing a string of terms including numerical coefficients and variables raised to whole-number exponents

radical equation

an equation containing at least one radical term where the variable is part of the radicand