Unit 2, Quadratic Functions: Day 7

Focus: Solving Quadratic Equations by Using Square Roots -- simplifying radicals.

Action Items:

1. Show me yesterday's homework 2.6.

2. Take notes on Solving Quadratic Equations using Square Roots.

3. Homework 2.7: WS #9.2B all.

Lesson Goal: At the end of this lesson, you will be able to

- write a radical in simplest form.

1. Link to prior learning (review) and preview:

Vocabulary: "solving" a quadratic means setting the equation equal to zero and finding the x-values that make the function zero.
- solutions are ;
- are where the graph crosses the
- solutions are also called zeros
- solutions are also called roots

We know how to solve a quadratic in Factored/Intercept Form: $LaTeX: y\:=\:a\left(x-p\right)\left(x-q\right)$ We set each factor equal to zero.

To solve Vertex Form and Standard Form (when it cannot be factored), we need radical skills.

2. Skills we need first: Simplifying Radicals!

What is a radical? The word radical comes from the Latin word radix which means root. The symbol looks like a check mark with a line extending from the top. The key parts of a radical expression are the index, root, and radicand:

A radical looks like $LaTeX: \sqrt{......}$ It matters if something is "inside" (or under) the radical or not, so you need to be careful when you write them.

$LaTeX: \sqrt{radicand}$ the radicand is the number under the radical

The index is the number "tucked in" to the "v" of the radical: $LaTeX: \sqrt[index]{radicand}\:\:$

Since square root is the most common radical, we don't write the two. For example, we just write $LaTeX: \sqrt{36}$ . We know $LaTeX: \sqrt{36}=\:6$ . The symbol is asking as for the "second root" of 36, or "what number times itself is 36." But you can have higher roots than just 2. For example $LaTeX: \sqrt[3]{8}$ is a cube root, or third root. The $LaTeX: \sqrt[3]{8}\:=\:2$ because $LaTeX: 2\cdot2\cdot2\:=\:8$ or $LaTeX: \sqrt[3]{2^3\:}\:=\:2$ .

Because quadratics involve we will focus on the square root.

Note: $LaTeX: \sqrt{x^2\:}=x$ This should seem obvious, but fully understanding this idea is central to the process of simplifying radicals.

Note: $LaTeX: \sqrt{x^2}\:=\:\sqrt{x}\:\cdot\sqrt{x}\:=\:x$ Some people call this "clearing the radical."

Examples:

$LaTeX: \sqrt{pizza}\cdot\sqrt{pizza}=\:\sqrt{\left(pizza\right)^2\:}\:=\:pizza$

$LaTeX: \sqrt{\Omega}\cdot\sqrt{\Omega}\:=\sqrt{\Omega^2}\:=\:\Omega$ It doesn't matter what the radicand is. If you have the $LaTeX: \sqrt{\left(radicand\right)^2}$ the result is just the radicand (what's under the radical).

When is a radical "simplified"?

when there are no perfect square factors under the radical
when there are no radicals in the denominator of a fraction

Examples:

$LaTeX: \sqrt{50\:}\:=\:\sqrt{25\cdot2}\:=\:\sqrt{25}\cdot\sqrt{2}=5\sqrt{2}$

$LaTeX: \sqrt{\frac{5}{4}\:}=\:\frac{\sqrt{5}}{2}$

$LaTeX: \frac{5}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}\:=\frac{5\sqrt{3}}{\sqrt{9}}\:=\:\frac{5\sqrt{3}}{3}$

We will do more examples in class. If you need more practice, check out Khan Academy or Patrick Math.

3. Homework 2.7: WS #9.2B: all. Use this reference page if you need a review of skills. If you need a further review of skills, use the resources!

Resources:

Quadratic Unit Homework Solutions

Reminder: all of the resources can be found in the Quadratics Resources Module.

Also, try Patrick JMT Math Tutorials: All About Quadratics (Complete Playlist)

And if you like Kahn Academy, use their Quadratics unit if you need more support.