Complex Numbers and Quadratic Equations

This is Unit Four of College Algebra: complex numbers and quadratic equations. The unit ends on a guarantee: every quadratic equation gives up its solutions. There is no such thing as one that cannot be solved. That is not true of most equation types, and it is worth knowing in advance, because it changes what going wrong means here. If the equation always surrenders, then any wrong answer is a handling failure. And the capstone names the real subject of the unit accordingly: custody. No root lost, none divided away, and none invented along the way. Four of the entries here are named errors, and every one is a custody failure. Two lose a solution that was there. One reports solutions that were never there. Watch for that pattern — it is what ties the unit together. The unit also introduces a new kind of number, and the two halves meet in the middle: quadratics that seemed to have no solutions turn out to have two, once the number system is large enough to hold them. By the end you will be able to do five things. One: divide complex numbers using the conjugate. Two: say why dividing an equation by its variable loses a solution. Three: say why a squared variable equal to a number has two answers. Four: read the discriminant before solving anything. Five: say why raising both sides to a power obliges a check.

Two entries. One definition, made because the previous number system could not supply a square root for a negative — and then the standard way of writing what results, which quietly contains every number you already had.

The definition.

Alg 181 The Imaginary Unit

Defined as the square root of negative one, so squaring it gives negative one back. Note that it is a number and not a variable — nothing is going to be solved for it.

The standard form.

Alg 182 Complex Numbers in Standard Form

Written as a real number plus a real multiple of the unit. Every ordinary number already fits, with the second part equal to zero — so this is an extension rather than a replacement.

Four operations. Adding and multiplying work by rules you already have, with one substitution applied at the end. Dividing needs something new — and it turns out to be the rationalising move from Unit One, pointed at a different problem.

Adding.

Alg 184 Adding and Subtracting Complex Numbers

Real parts combine with real parts and imaginary parts with imaginary parts, exactly the way like terms combine.

Multiplying.

Alg 185 Multiplying Complex Numbers

Distribute as with any two-term expressions, and then replace the squared unit with negative one wherever it appears. That replacement is the step that makes the answer come out real in places you would not expect.

The tool for dividing.

Alg 187 The Complex Conjugate

Flip the sign on the imaginary part, and multiplying the two together gives a plain real number — the cross terms cancel and the squared unit turns the rest real.

Watch division run.

Alg 188 Dividing Complex Numbers

Multiply top and bottom by the bottom's conjugate, which turns the denominator into a real number and leaves the value untouched. Only the denominator gets conjugated — the numerator is carried along unchanged.

What the new number was built for, with one instruction about order. Then a pattern in its powers that repeats every four steps, which turns any power into a small remainder problem.

The original purpose.

Alg 190 Square Roots of Negative Numbers

The root of a negative number becomes the unit times the root of the positive part. Extract the negative one before doing anything else — the radical rules from Unit One assume a nonnegative inside, and applying them first produces contradictions.

The cycle.

Alg 191 The Powers of i

The powers run through four values and repeat forever. Any exponent reduces to its remainder on division by four, which turns an enormous power into one of four answers.

The unit's main object, and the property every solving method depends on. That property has one condition attached, and it is the condition students most often ignore — because ignoring it produces an answer that looks perfectly reasonable.

The object.

Alg 196 Quadratic Equations

An equation whose highest power is two, written with everything on one side. Every method in the unit reads from that arrangement, so it comes first.

The property.

Alg 197 The Zero-Product Property

If a product equals zero, at least one of the factors is zero. The condition is the zero — a product equal to six tells you nothing about the factors, because six has many factorisations.

The first method.

Alg 198 Solving a Quadratic by Factoring

Move everything to one side, factor completely, and set each factor to zero. The right side must actually be zero before the property applies — with anything else there, splitting into cases is not licensed by anything.

Four entries, and they are the heart of the unit's custody theme. Two ways of ending up with half an answer, each paired with the error entry naming it. Both come from a move that looks like simplification and is actually destruction.

The first temptation.

Alg 200 Keeping Every Factor

With the variable on both sides of a product, factor rather than divide. Each factor carries a solution, and cancelling one throws its solution away with it.

The error.

Alg 201 The Divided-Away Solution

Dividing both sides by the variable quietly assumes the variable is not zero — and zero is very often one of the answers. The step is invisible because nothing about it looks like an assumption.

The second place roots hide.

Alg 203 The Square Root Property

A squared variable equal to a number has two solutions, positive and negative. Note where the sign belongs — it is a property of the equation, not something the radical symbol does.

And its error.

Alg 204 The Lost-Root Error

Reporting only the positive root. Unit One established that the radical returns the nonnegative value, and that convention governs the symbol — the equation has two answers regardless of what the symbol returns.

A technique that works on any quadratic, and then the formula, which is that technique performed once with letters so nobody has to perform it again. And a single number inside it that reports what the answers will be before you compute them.

The technique.

Alg 206 Completing the Square

Add the square of half the middle coefficient to both sides, which turns the variable side into a perfect square that can be unpacked. Both sides receive it — adding to one side alone produces a different equation.

The formula.

Alg 208 The Quadratic Formula

A single expression solving any quadratic from its three coefficients. It is the previous technique carried out on the general case — which is why it always works, and why it is not a separate idea.

The number inside it.

Alg 209 The Discriminant

The expression under the radical, and its sign tells you what kind of answers are coming before any of the arithmetic is done — two, one, or a pair of a different kind entirely.

The formula, worked.

Alg 211 Solving with the Quadratic Formula

Take the three coefficients, evaluate the discriminant, then finish with both signs. Watch the sign of the constant — it travels into the discriminant, and a negative constant makes that subtraction into an addition.

One entry, and it is where the unit's two halves meet. Equations that had no solutions when the numbers stopped at the real line turn out to have two, and they arrive in a specific relationship.

The meeting point.

Alg 213 Complex Solutions of Quadratics

A negative discriminant gives two solutions in the new number system, and they are conjugates of one another. They never appear alone — finding one means the other is there.

A different equation type, and the other half of the custody problem. The previous topics lost solutions. This method invents them — and the invention is a property of the method rather than a mistake in performing it.

The type, and the move.

Alg 216 Radical Equations

The variable sitting inside a radical. Get the radical alone, then raise both sides to whatever power undoes it.

What raising does.

Alg 217 Checking After Raising to a Power

Raising both sides to a power can produce values solving the new equation and failing the original. This is a property of squaring, which loses sign information — so the check is part of the method rather than a precaution.

The error.

Alg 218 The Unchecked-Squaring Error

Reporting every root of the raised equation as an answer. The fabricated ones are not hard to detect — they fail the original immediately — and they survive only because nobody substituted them back.

The procedure.

Alg 220 Solving a Radical Equation

Get the radical alone, raise both sides, solve what remains, and test every candidate against the original. A candidate that fails is discarded without apology — the checking step is the method working, not a sign anything went wrong.

Three shapes that look new and are not. Each one reduces to something already handled — by a substitution, by an exponent, or by splitting into cases — and the last one has an outcome the others do not.

The first.

Alg 222 Equations Quadratic in Form

When one expression repeats, replacing it with a single letter turns the equation into a quadratic. Solve, then translate back — the substituted letter is scaffolding and never appears in the answer.

The second.

Alg 224 Equations with Rational Exponents

A variable carrying a fractional exponent: raise both sides by the flipped version of it, returning the variable to the first power. And where the fraction's bottom is even, the checking audit from two topics back comes with it.

And the third.

Alg 226 Absolute Value Equations

An absolute value set equal to a positive number splits into two equations. Set equal to a negative number it has no solutions at all — distance is never negative, so nothing satisfies it.

The unit closes on the guarantee the opening quoted, and then names what the unit was really about — which is not the solving at all.

The capstone. A guarantee, and a subject.

Alg 230 Every Quadratic Surrenders

Every quadratic gives up its solutions — there is no unsolvable one. So the unit's real subject was custody: keeping every root that exists, refusing to divide any away, and rejecting the ones the methods manufacture.

Five things you can now do. You can divide complex numbers by multiplying through by the denominator's conjugate — Unit One's rationalising move, pointed at a different problem. You can say why dividing an equation by its variable loses a solution: the step assumes the variable is not zero, and zero is often one of the answers. You can say why a squared variable equal to a number has two solutions, and that the radical's convention governs the symbol rather than the equation. You can read the discriminant before solving, and you know a negative one gives two solutions rather than none — they arrive as a conjugate pair. And you can say why raising both sides obliges a check: squaring loses sign information, so the check is the method working rather than a precaution.