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# Formulas

Formulas are frequently one of the first things considered when people think about mathematics. Formulas are equations that relate two or more quantities and often have a practical purpose rather than feeling abstract. Formulas are a constant throughout all levels of mathematics, and you'll be asked to memorize them, understand when to apply them, and handle the number-crunching that goes into each use case.

One of the first formulas that most students encounter is the perimeter of a square. Since all four of a square's sides are the same length, you can calculate its perimeter by multiplying the length of one side by four. It's expressed like this:

$P=4s$ , where $P$ is the perimeter and $s$ is the length of one side.

Other shapes have different perimeter formulas. For example, here is the formula for the perimeter of a rectangle:

$P=2L+2W$ , where $P$ is the perimeter, $L$ is the Length, and $W$ is the Width.

These basic formulas illustrate many of the constants that hold for all formulas, namely that you need to know which one to apply in any given situation and how to perform the operations involved. Formulas aren't limited to geometry either, as many relate to algebra or the sciences. In fact, professionals in fields such as engineering and finance use formulas all of the time.

## Applying a formula

The most basic way of using a formula is to apply it by solving an equation. For example, the quadratic formula is a defining feature of Algebra I courses and can be used to solve for x whenever you're faced with a quadratic equation in the following format:

$0=a{x}^{2}+bx+c$

Many students get intimidated when they see the quadratic formula for the first time though because it looks extremely complicated:

$x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$

The first step toward using this monster is to make sure you understand it. The expression under the square root sign is called the discriminant and determines whether you'll get two answers (if it's positive), zero answers without complex numbers (if it's negative), or one answer (if it's zero). The sign acts as both a + and a -, which is why you get two answers with a positive discriminant.

Now that you understand what the quadratic formula is asking you to do, all you have to do is plug in your numbers for a, b, and c and work out the answer. Importantly, your answer may involve radicals if the solution isn't a perfect square, so bear that in mind as you double-check your work.

## Useful formulas to know

Formulas are first introduced in elementary school and continue to come up in college and beyond, so an exhaustive list would be impossible. That said, the chart below includes some useful formulas for students of all levels of mathematics:

 Formula Purpose ${a}^{2}$ Area of a square $2\pi \left(r\right)$ Circumference of a circle $y={a}^{2}+{b}^{2}={c}^{2}$ Pythagorean Theorem $y=mx+b$ Slope Intercept $\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ Slope $\sqrt{\left({{x}_{1}}^{2}-{{x}_{2}}^{2}\right)+\left({{y}_{1}}^{2}-{{y}_{2}}^{2}\right)}$ Distance Formula (between two points) $\frac{1}{2}×\mathrm{base}×\mathrm{height}$ Area of a triangle ${r}^{2}={\left(x-h\right)}^{2}+{\left(y-k\right)}^{2}$ Equation of a circle $A=P{\left(1+\frac{R}{N}\right)}^{nt}$ Compound Interest $y={x}^{2}$ Parabola opening up

This chart draws examples from multiple levels of mathematics, so it's okay if you don't understand what some of them mean. Of course, you could also use this chart to get a sneak peek at the types of math you might see in more advanced courses!

## Using one formula to determine another

All formulas are algebraic equations at their core, allowing you to use your algebra skills to manipulate a formula you know into the one you need. Let's look at the formula for converting temperature from Celsius units to Fahrenheit as an example:

$F=\frac{9}{5}C+32$

With this formula, you can simply plug in your numbers and convert any Celsius temperature to Fahrenheit. But what if you have the Fahrenheit temperature and are looking for Celsius? In that case, you need to solve for C on one side of the equation by treating all of the other variables (only F in this case) as numbers and operating on both sides.

First, subtract 32 from both sides:

$F-32=\frac{9}{5}C$
Next, divide each side of the equation by $\frac{9}{5}$ (which is the same as multiplying by $\frac{5}{9}$ )
$\frac{5}{9}\left(F-32\right)=C$

Finally, simplify the resulting expression to get the formula for converting Fahrenheit temperatures to Celsius:

$C=\frac{5}{9}F-\frac{160}{9}$

Voila! We have successfully determined the formula for converting Fahrenheit temperatures to Celsius from the formula converting Celsius to Fahrenheit. Many other formulas can be reverse-engineered in this manner, potentially reducing the need for rote memorization if a related formula is provided to you.

## Formula practice questions

a. What is the perimeter of a square with one side measuring 6 inches?

$\mathrm{Perimeter}=4×\mathrm{side length}$

$\mathrm{Perimeter}=4×6\phantom{\rule{2pt}{0ex}}\mathrm{inches}=24\mathrm{inches}$

b. What is the area of a square with one side measuring 6 inches?

$\mathrm{Area}={\mathrm{side length}}^{2}$

$\mathrm{Area}=6\phantom{\rule{2pt}{0ex}}\mathrm{inches}×6\phantom{\rule{2pt}{0ex}}\mathrm{inches}=36\phantom{\rule{2pt}{0ex}}\mathrm{square inches}$

c. What is the perimeter of a rectangle with a length of 10 inches and a width of 12 inches?

$\mathrm{Perimeter}=2×\left(\mathrm{length}+\mathrm{width}\right)$

d. Solve for x: $5{x}^{2}-17x+12=0$

$5{x}^{2}-17x+12=0$

e. What is the circumference of a circle with a radius of 5 meters? Use 3.14 for the value of 𝛑.

$\mathrm{Circumference}=2×\pi ×\mathrm{radius}$

f. What is the slope of the following equation? $y=-\frac{1}{3}x+7$

$-\frac{1}{3}$

g. If you invested \$1,000 at a 9% interest rate compounded monthly, how much money would you have after 18 months?

$\mathrm{Future Value}=\mathrm{Principal}×{\left(1+\frac{\mathrm{rate}}{n}\right)}^{nt}$

$\mathrm{Principal}=1000$

$\mathrm{Rate}=9%=0.09$

$\text{Number of compounding periods per year (n)}=12$

$\mathrm{Future Value}=1000×{\left(1+\frac{0.09}{12}\right)}^{12×1.5}$

$\mathrm{Future Value}=1000×{\left(1.0075\right)}^{18}$

$\mathrm{Future Value}\approx 1143.96$

h. The formula for determining the area of a trapezoid is $A=\frac{1}{2}h\left({b}_{1}+{b}_{2}\right)$ , where $h$ is the height and ${b}_{1}$ and ${b}_{2}$ are the base lengths. What formula could you use to find the height of a trapezoid by solving for $h$ ?

$2A=h\left({b}_{1}+{b}_{2}\right)$

To solve for h, first multiply both sides by 2:

$2A=h\left({b}_{1}+{b}_{2}\right)$

Then divide both sides by $\left({b}_{1}+{b}_{2}\right)$ :

$h=\frac{2A}{{b}_{1}+{b}_{2}}$

## Flashcards covering the Formulas

Algebra 1 Flashcards

## Get help with formulas today

Working with formulas involves operational math, rote memorization, and a fundamental understanding of the concepts behind it. If your student is struggling to choose the correct formula for the situation or remember what all of those variables are supposed to indicate, a 1-on-1 tutor can help them work toward overcoming their formula-related learning obstacles. Contact the friendly Educational Directors at Varsity Tutors right now to learn more about all of the benefits of private instruction or to sign up.

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