1
Pick a method deliberately
Use factoring when it is quick, the square root property when there is no linear term, and the formula when nothing else applies — not always the formula.
Course · Algebra
Quadratics and Complex Numbers
One equation shape, four solution methods, and a number system that has to be extended so that every one of them always finishes. Completing the square is the centre of the course: it derives the formula and it puts a parabola in vertex form.
ab = 0 ⟹ a = 0 or b = 0
x = (−b ± sqrt(b² − 4ac)) / 2a
b² − 4ac > 0, = 0, < 0 two, one, none (real)
i² = −1 y = a(x − h)² + k
What this course is about
- Subject
- Algebra
- Lessons
- 14
- Format
- Interactive
Overview
What you will be able to do
By the end you can choose and carry out a solution method, classify the roots before solving, express non-real roots in complex form, and recover a parabola's decisive features from its equation.
2
Complete the square
Carry out the process on any quadratic, and recognise it as both the derivation of the formula and the route to vertex form.
3
Use the discriminant
Predict the number and kind of solutions before solving, and connect that to the number of times the parabola meets the x-axis.
4
Work with complex numbers
Add, multiply and divide complex numbers, use the conjugate, and interpret a conjugate pair of roots as a parabola that misses the axis.
Lessons
14 lessons
Quadratics are solved and their solutions counted first; the complex numbers are built next; and last they are graphed, optimised and extended.
Quadratic Equations and the Zero Product PropertyClassify a quadratic and split only a product equal to zero.Not marked
Solving by FactoringRecognise when factoring is efficient, then find and check every root.Not marked
The Square Root PropertyIsolate a square, take both roots and simplify them exactly.Not marked
Completing the SquareComplete a square in exact arithmetic and solve the resulting equation.Not marked
The Quadratic FormulaDerive the quadratic formula, substitute signed coefficients and reduce both roots.Not marked
The DiscriminantUse the discriminant and equation shape to predict roots and choose a method.Not marked
Complex NumbersConvert negative square roots to i and identify both parts of a complex number.Not marked
Operations with Complex NumbersCompute in a + bi form and use a conjugate to divide.Not marked
Complex Roots of QuadraticsCarry a negative discriminant through the formula and verify the conjugate pair.Not marked
Graphs of Quadratic FunctionsRecover a parabola's direction, intercepts, axis and vertex from its equation.Not marked
Vertex Form and the Axis of SymmetryConvert to vertex form, read the extreme point and verify it independently.Not marked
Maximum and Minimum ProblemsBuild a quadratic model, enforce its domain and report the attainable optimum.Not marked
Quadratic InequalitiesPartition the real line by the roots and justify every included interval.Not marked
Equations Reducible to Quadratic FormRecognise a repeated inner expression, solve in u and return every valid x.Not marked
Recommended background
Factoring and radical simplification.
Practice suggestions
On each lesson, cover the complete worked example before attempting the faded rehearsal beneath it. The rehearsal supplies the first strategic decision but leaves the algebra and check to you; only then use the quiz as independent retrieval.
Complete the square by hand at least five times. It is the one procedure in this course that pays for itself twice — once as the formula, once as vertex form.
Use the discriminant before you solve, every time. Knowing the answer is a conjugate pair before you start prevents the most common error, which is discarding a negative under the root.
In the graphing lessons, change one coefficient at a time and watch. a, h and k each do exactly one thing, and seeing them separated is worth more than the formula.
What it does not cover
Polar form of complex numbers, De Moivre's theorem and complex roots of unity, which need trigonometry.
Cubic and quartic formulas. Polynomials and Factoring's rational root theorem is the tool this path gives for higher degrees.
Conic sections in general. The parabola appears here as the graph of a quadratic function, not as a locus.