You own a company selling apps and you need to know the best selling price for maximum profit. If this quadratic equation could help you find the answer...

...would you be able to solve it?

Solving quadratic equations is all about finding the "x"!

## But First...Some Need to Know Terms!

### What is a quadratic equation?

Quadratic comes from the word "quad", which means "square". So, a quadratic function is a **polynomial function of a degree of 2**. "Polynomial" means the sum of many terms: "poly" (many) and "nomial" (name).

When plotted on a graph, quadratic equations are **U-shaped** (like a horseshoe) and can open upwards (as in a smiley face) or downwards (as in a sad face).

### What are factors?

Factors **multiply **to get a number.

Factoring is the inverse (or **reverse action**) of multiplying. For example, when **factoring 18**, you'll find **numbers that multiply** to get 18 (1, 2, 3, 6, 9, 18). This is where knowing multiplication tables comes in handy!

### What is standard form?

Standard formin math simplifies rules for clear understanding, communication, and computation. Quadratic equations are easier to factor in standard form...

...where "a", "b", and "c" are numbers and **"x" is the variable (or unknown).**

"a" and "b" are called

**coefficients**— a constant number multiplied by a variable."c" is called a

**constant**— it's a fixed value not multiplied by a variable.

If the coefficient is absent, you can understand it to be "1". If "a" = 1, you only need to find the factors of "c". An example of handling a quadratic when "a" > 1 will be provided later.

### How are quadratic equations made?

Multiplying binomials is** similar to multiplying two-digit numbers**. The exception is **two algebraic expressions are being multiplied** instead of only digits.

The video below provides an example:

## Method #1: Factoring

Factoring is the inverse (or reverse operation) of multiplying two algebraic expressions.

### Example #1

You have the quadratic equation:

To factor this, you need to identify the algebraic expressions that were **multiplied together**. Your **end result** will look like this...

...where "x" is the **variable (x that makes x squared) **in the equation and "#" will be the **factors of 4 (the numbers that multiply to get 4)**

The video below explains factoring quadratics a bit further.

One method to finding those factors is using a table:

Label a table with the headings **Factor, Added, Subtracted**. Use the abbreviations "F", "A", and "S".

*Image created by Wendy McMillan, 2016*

**List**factor pairs (of 4) in the**Factors (F)**column**Add**factor pairs in the**Added (A)**column**Subtract**factor pairs in the**Subtracted (S)**columnFind the

**answer**under heading A or S**for the middle term (3)**.

Answer:

The **sign of the middle number** determines positive or negative constants. Since the middle number 3 is positive, 4 must be positive (think 4 - 1 vs. 1 - 4).

### Example #2

When the value "a" > 1, you'll need to multiply the "a" and "c" values together, then factor.

Take a look at this equation:

**Multiply**6 ("a") x 6 ("c") = 36**Factor**36You'll see that

**factors 9 and 4 when subtracted total 5**

With "a" and "c" values as 6, **further factorize** 9 and 4 to form the required factor groups [2 x 2 = 4 and 3 x 3 = 9 so 9 - 4 = 5 and 9 x 4 = 36.]

## Find "x" by Solving Algebraic Equations

Now that you have your factors, you can find the **value of "x"**.

Set

**each factor group equal to 0**(zero)Solve for "x" using the

**order of operations**

### Example #1:

### Example #2:

## Method #2: Complete the Square

What if you have an algebraic equation like this?

There **aren't any factors of 3** that add or subtract to get 6!

You'll need to **complete the square** — meaning, you'll need to **make the formula a perfect square**.

### You'll do this by:

**Rearranging**the equation with variables on one side

**Dividing**the "b" term (6) by 2

**Squaring**that answer

**Adding**that answer to both sides of the equation

**Factor**the new equation

This equation could

**also be written like this**...

...which is why it's called **a perfect square**!

## Method #3: Use the Square Root Property

What about quadratic equations in this form?

The middle term ("bx") is missing!

You'll need the **square root method** to solve these types of equations.

**Square roots** are the inverse or **reverse action of squaring a number**.

First,

**isolate "x"**by eliminating any coefficients or additive/subtractive terms by**adding**3 to both sides, then**simplifying**

**Divide**each term by 2

**End**with this equation

Take the

**square root of both sides**

This is where knowing **squared numbers** comes in handy!

Your result should have **a positive AND a negative answer** since a negative times negative number and a positive times a positive number **both equal a positive answer**.

## Method #4: Use the Quadratic Formula

**Some equations are rebellious** and just can't be solved using any of the methods above.

### Example:

When the value "a" > 1, multiply the "a" and "c" values, then factor!

**3 x 7 = 21**....but there are **NO factors of 21 (1, 3, 7, 21)** that add or subtract **to get 5**.

### Enter the quadratic formula!

*GIF created by Wendy McMillian, 2023 with PowerPoint and pngtree.com*

In standard equations "a", "b", and "c" will be **numerical values**. That is:

a = 3

b = 5

c = -7 (don't forget to carry the negative along)

**Replace each variable** in the quadratic formula **with the corresponding number** minding all negatives and positives. Use **parenthesis **to keep things **organized**!

**Begin solving** under the radical:

**End** by simplifying:

If you enter these equations into a calculator you end up with nasty-looking decimals! It's better to just leave it as a fraction.

## Method #5: Graph to Solve the Algebraic Equation and Find "x"

One easy way to solve quadratics is by graphing! Remember the equation at the beginning of this lesson?

**No factors **of 458 (1, 2, 229, 458) **add or subtract** to get **72**.

These **big numbers** **will cause major headaches **too when used in the quadratic formula!

### Desmos will save the day!

Desmos is a graphing calculatorthat handles big equations.

**Type the equation**into Desmos**Zoom out**using the "-" buttonFind where

**the graph crosses the x-axis (2 places)**The "x"-values

**are the factors**for the equation

At the beginning of this lesson, you were asked if you could find the best price to sell apps and make a maximum profit.

*Graphing videos created by Wendy McMillian using Desmos graphing calculator online and Screencastify*

Find the answer by **scrolling to the top of the graph to the x-value** at the graph's peak — which is 36.

You'll need to **sell 36 apps** in order to make a **profit**!

## Take Action

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