Welcome back to class! Today, we are shifting our focus back to quadratic functions, but with a specific twist. We are looking at equations that contain an $x^2$ term but are missing the linear $x$ term (where $b=0$). Because there is only one variable term to worry about, we can use a very powerful and direct tool: The Square Root Method.

The Square Root Property

The core concept for today is simple but requires close attention to detail. The Square Root Property states that to solve a quadratic equation in the form $x^2 = a$, you take the square root of both sides.

$$ \text{If } x^2 = a \text{ and } a \text{ is a positive real number, then } x = \pm\sqrt{a} $$

The "Checkmark" Grows Up

You might notice the title of this post mentions the "checkmark." The radical symbol ($\sqrt{\quad}$) looks a bit like a checkmark. However, as we "grow up" in our mathematical maturity, we have to remember a crucial rule that is often overlooked: equations with $x^2$ usually have two solutions.

When you take the square root of a number to solve for $x$, you must include both the positive and the negative root (indicated by the $\pm$ symbol). For example:

  • If $x^2 = 169$, then $x = \pm 13$. The solutions are $13$ and $-13$.

Step-by-Step Guide

  1. Isolate the Squared Term: Use inverse operations (adding, subtracting, multiplying, or dividing) to get $x^2$ (or the squared parenthesis) by itself on one side of the equal sign.
  2. Apply the Square Root: Take the square root of both sides of the equation.
  3. Don't Forget the $\pm$: meaningful solutions usually come in pairs!
  4. Simplify: If the number isn't a perfect square, simplify the radical (e.g., $\sqrt{12} = 2\sqrt{3}$) or round to the nearest hundredth if the instructions ask for it.

Special Cases

As you work through your notes, keep an eye out for these special scenarios:

  • No Real Solution: If you isolate the variable and get $x^2 = -49$, you cannot take the square root of a negative number in the real number system. There is no real solution.
  • Falling Objects: In the attached notes, we see the physics formula $h = -16t^2 + h_0$. This is a perfect application of the square root method to find out how long it takes for an object to hit the ground!

Class Assignments & Resources

  • Class Notes: Review the attached slides for examples on isolating $x^2$ and simplifying radicals.
  • Homework: Practice B Worksheet, Problems #10-18 (all).
  • Discussion Question: Now that we have looked at the square root method, do you think it is easier or harder than solving by factoring (when $b=0$)? Explain your reasoning. Are there problems in your homework that could not be solved with factoring but could be solved with this method?