The Quadratic Formula: Your Universal Solver

Welcome to the final new concept of this cycle: the quadratic formula! This powerful tool allows us to find the solutions (also known as roots or zeros) to any quadratic equation, regardless of whether it can be factored or solved using square roots. No matter the equation, if you can write it in standard form, you can solve it with the quadratic formula.

Standard Form and the Quadratic Formula

Remember that the standard form of a quadratic equation is: $$ax^2 + bx + c = 0$$

Where $a$, $b$, and $c$ are coefficients. The quadratic formula itself is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Why Use the Quadratic Formula?

  • Universality: It works for all quadratic equations.
  • No Factoring Required: Avoid the trial and error of factoring.
  • Handles All Solutions: Finds real and complex solutions.

The Discriminant: Unveiling the Nature of Solutions

The expression under the square root in the quadratic formula, $b^2 - 4ac$, is called the discriminant. The discriminant tells us about the nature of the solutions:

  • If $b^2 - 4ac > 0$: Two distinct real solutions.
  • If $b^2 - 4ac = 0$: One real solution (a repeated root).
  • If $b^2 - 4ac < 0$: Two complex solutions.

Example: Putting the Formula to Work

Let's solve the quadratic equation $2x^2 - 16x + 27 = 0$ using the quadratic formula.

  1. Identify a, b, and c: In this case, $a = 2$, $b = -16$, and $c = 27$.
  2. Substitute: Plug these values into the quadratic formula:
    3. $$x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(2)(27)}}{2(2)}$$
  3. Simplify: $$x = \frac{16 \pm \sqrt{256 - 216}}{4}$$ $$x = \frac{16 \pm \sqrt{40}}{4}$$ $$x = \frac{16 \pm 2\sqrt{10}}{4}$$ $$x = 4 \pm \frac{\sqrt{10}}{2}$$

Therefore, the solutions are $x = 4 + \frac{\sqrt{10}}{2}$ and $x = 4 - \frac{\sqrt{10}}{2}$.

Class Resources

  • Class Notes: Review the attached PDFs for detailed explanations and examples.
  • Class Assignments: Work through the assigned problems to solidify your understanding.
  • Homework: Complete problems on Pg. 105 #18-29 for practice.

Discussion Question

The quadratic formula is incredibly useful in many real-world applications. For today's discussion, find a website that either allows you to plug in the values of $a$, $b$, and $c$ to compute the solution, or directly solves the quadratic equation for you. These tools are great for checking your work! However, remember that understanding the process is crucial. Avoid relying solely on these websites, as they may not always show the steps or provide the solutions in the desired form (e.g., they might give complex solutions when you are looking for real solutions).

Good luck, and remember, practice makes perfect!