Mastering the Quadratic Formula: Khan Academy Examples

Welcome to Professor Baker's Math Class! In this post, we'll delve into the quadratic formula, a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$. We'll be using examples inspired by the Khan Academy's excellent resources to guide you through the process. Whether you're just starting out or need a refresher, these examples will help you build confidence and master this important concept.

What is the Quadratic Formula?

The quadratic formula is given by:

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

Where $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$. The term $b^2 - 4ac$ is called the discriminant, and it tells us about the nature of the roots:

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

Example 1: Solving a Basic Quadratic Equation

Let's consider the equation $x^2 + 5x + 6 = 0$. Here, $a = 1$, $b = 5$, and $c = 6$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-5 \pm \sqrt{5^2 - 4(1)(6)}}{2(1)}$$ $$x = \frac{-5 \pm \sqrt{25 - 24}}{2}$$ $$x = \frac{-5 \pm \sqrt{1}}{2}$$ $$x = \frac{-5 \pm 1}{2}$$

So, the two solutions are:

  • $x = \frac{-5 + 1}{2} = -2$
  • $x = \frac{-5 - 1}{2} = -3$

Example 2: Dealing with a Negative Discriminant

Now, let's look at $x^2 + 2x + 5 = 0$. Here, $a = 1$, $b = 2$, and $c = 5$. Applying the quadratic formula:

$$x = \frac{-2 \pm \sqrt{2^2 - 4(1)(5)}}{2(1)}$$ $$x = \frac{-2 \pm \sqrt{4 - 20}}{2}$$ $$x = \frac{-2 \pm \sqrt{-16}}{2}$$

Since the discriminant is negative, we have complex roots:

$$x = \frac{-2 \pm 4i}{2}$$ $$x = -1 \pm 2i$$

The solutions are $x = -1 + 2i$ and $x = -1 - 2i$.

Tips for Success

  • Identify a, b, and c correctly: This is the crucial first step.
  • Simplify carefully: Pay attention to signs and order of operations.
  • Check your answers: Substitute your solutions back into the original equation to verify they are correct.

The quadratic formula can seem daunting at first, but with practice and a clear understanding of the steps involved, you can confidently solve a wide range of quadratic equations. Keep practicing, and don't hesitate to review the Khan Academy resources for more examples and explanations. Good luck!