Welcome to Sections 1.3 - 1.5

Today's class marked a significant shift as we expanded our toolkit beyond real numbers and dove deep into quadratic equations. We covered a lot of ground in Sections 1.3 through 1.5, moving from the definition of imaginary units to the application of the quadratic formula.

Section 1.3: Complex Numbers

We began by defining the imaginary unit, $i$, where $i = \sqrt{-1}$ and consequently $i^2 = -1$. This allows us to work with Complex Numbers in the standard form $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

Key Operations:

  • Addition/Subtraction: Combine like terms (real with real, imaginary with imaginary).
  • Multiplication: Use the FOIL method and remember that $i^2$ becomes $-1$.
  • Division: To simplify a fraction with a complex number in the denominator, multiply both the numerator and denominator by the complex conjugate of the denominator. If the denominator is $a + bi$, the conjugate is $a - bi$.

Section 1.4: Solving Quadratic Equations

Next, we tackled quadratic equations in the general form $$ax^2 + bx + c = 0$$. We discussed four distinct methods to solve these, and it is crucial to know when to apply each one:

  1. Factoring: The fastest method if the equation is easily factorable. We use the Zero Product Property: if $A \cdot B = 0$, then $A=0$ or $B=0$.
  2. Square Root Property: Best used when there is no $x$ term (i.e., $(expression)^2 = k$). Remember to include the $\pm$ symbol: $$x = \pm\sqrt{k}$$
  3. Completing the Square: A powerful technique that forces a perfect square trinomial. This is the method used to derive the quadratic formula.
  4. The Quadratic Formula: The universal method that works for any quadratic equation: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Section 1.5: Analyzing Solutions

Finally, we looked at how to predict the type of solutions an equation will have by looking at the discriminant, which is the part of the quadratic formula under the radical: $b^2 - 4ac$.

  • If $b^2 - 4ac > 0$, there are two distinct real solutions.
  • If $b^2 - 4ac = 0$, there is exactly one real solution (a repeated root).
  • If $b^2 - 4ac < 0$, there are two complex (imaginary) solutions.

Please review the attached class notes and video to see step-by-step examples of these problems. Practice is key to mastering the arithmetic of complex numbers and memorizing the quadratic formula!