Quadratics & Complex Numbers

The skills, in order

Each skill builds on the ones before it.

1. 1

   Factor quadratic expressions

   Factor quadratic expressions $ax^2+bx+c$, including a greatest common factor, $a \neq 1$, difference of squares, and perfect-square trinomials.

2. 2

   Solve quadratic equations by factoring

   Solve quadratic equations by rewriting in standard form, factoring, and applying the zero-product property.

3. 3

   Solve quadratic equations using square roots

   Solve equations of the form $ax^2+c=0$ and $a(x-h)^2=k$ with real solutions by isolating the square and taking square roots.

4. 4

   Complete the square to write vertex form

   Rewrite $ax^2+bx+c$ in the form $a(x-h)^2+k$ by completing the square.

5. 5

   Solve quadratic equations by completing the square

   Solve quadratic equations with real solutions by completing the square and taking square roots.

6. 6

   The imaginary unit and square roots of negative numbers

   Use $i=\sqrt{-1}$ and $i^2=-1$ to write square roots of negative numbers as $bi$, simplify powers of $i$, and identify the real and imaginary parts of $a+bi$.

7. 7

   Add, subtract, and multiply complex numbers

   Add, subtract, and multiply complex numbers and write the result in the form $a+bi$.

8. 8

   Complex conjugates and division

   Find the conjugate of a complex number and divide complex numbers by multiplying by the conjugate of the denominator, writing the result as $a+bi$.

9. 9

   Use the discriminant to classify solutions

   Compute $b^2-4ac$ and use it to determine whether a quadratic equation has two real, one real, or two complex (non-real) solutions.

10. 10

    Solve quadratics with the quadratic formula

    Solve any quadratic equation with real coefficients using the quadratic formula, giving real or complex solutions in simplest form.

11. 11

    Key features of quadratic functions and graphs

    Identify the vertex, axis of symmetry, maximum or minimum, intercepts, and direction of opening of a quadratic function from its equation in standard, vertex, or factored form, and from its graph.

12. 12

    Model and solve problems with quadratics

    Write and solve quadratic equations for real situations (projectile motion, area, revenue) and interpret the vertex and roots in context.