Unit 2 of 8

Choosing Useful Algebra

  • 10 lessons
  • Parts 15 to 34

Much of the work in a calculus problem is algebra, and the hard part is choosing which rewrite helps. This unit practices the moves that come up most often, expanding and factoring, cancelling common factors without losing excluded values, combining fractions and handling powers and roots with the right signs.

The second half solves equations and inequalities and checks the answers, rationalizes expressions with conjugates, completes the square, uses the quadratic formula and rearranges a formula to make a chosen variable the subject.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Rewriting Expressions

  1. Expanding and Factoring for a Purpose

    How to tell terms from factors, expand a product or a square without losing the cross terms, factor a difference of squares completely over the reals, and check each rewrite by reversing it.

    Theory part 15Practice part 16

  2. Cancellation and Excluded Values

    Which factors you may cancel in a fraction, why the inputs that made the original denominator zero stay excluded afterwards, and how a zero numerator differs from a zero denominator.

    Theory part 17Practice part 18

  3. Common Denominators and Complex Fractions

    How to add and subtract algebraic fractions over a common denominator, how to flatten a fraction inside a fraction, and how to keep every original restriction while doing it.

    Theory part 19Practice part 20

  4. Powers, Roots, and Signs

    What negative and fractional exponents mean, why a square root is never negative, why the square root of x squared is the absolute value of x, and where a minus sign really belongs.

    Theory part 21Practice part 22

Solving Equations and Inequalities

  1. Solving Equations and Checking Solutions

    Which steps keep an equation's solutions exactly the same, which steps can add or lose solutions, and why every candidate from a radical or rational equation is checked in the original equation.

    Theory part 23Practice part 24

  2. Solving Inequalities on a Number Line

    How to solve linear and compound inequalities step by step, why multiplying by a negative number flips the sign, and how to write the exact solution set as an interval.

    Theory part 25Practice part 26

Radicals and Quadratics

  1. Conjugates and Rationalizing Expressions

    How multiplying by a conjugate clears a square root from a numerator or denominator, and how to keep track of the original domain so the new form is equal to the old one.

    Theory part 27Practice part 28

  2. Completing the Square

    How to rewrite a quadratic as a constant multiple of a square plus a constant without changing its value, and how that form shows the smallest or largest value at a glance.

    Theory part 29Practice part 30

  3. The Quadratic Formula and Real Solutions

    How to read off the coefficients of a quadratic equation, apply the quadratic formula correctly, and use the discriminant to tell in advance whether there are two, one or no real solutions.

    Theory part 31Practice part 32

Rearranging Formulas

  1. Making a Variable the Subject

    How to isolate one variable in a formula with inverse operations, which nonzero and sign conditions the result needs, and how to read the new formula in context.

    Theory part 33Practice part 34