## Square Roots Odd And Even:

There are 2 possible roots for any positive real number. A positive root and a negative root. Given a number*x*, the square root of *x* is a number *a* such that*a2 = x*. Square roots is a specialized form of our commonroots calculator.

“Note that any positive real number has two square roots, one positive and one negative. For example, the square roots of 9 are -3 and +3, since 2 = 2 = 9. Any nonnegative real number*x* has a unique nonnegative square root r this is called the principal square root ………. For example, the principal square root of 9 is sqrt = +3, while the other square root of 9 is -sqrt = -3. In common usage, unless otherwise specified, “the” square root is generally taken to mean the principal square root.”.

## Completing The Square When B = 0

When you do not have an x term because b is 0, you will have a easier equation to solve and only need to solve for the squared term.

For example: Solution by completing the square for:

Eliminate b term with 0 to get:

Keep \ terms on the left and move the constant to the right side by adding it on both sides

Take the square root of both sides

## Numbers 0 Through 10 Cubed And The Resulting Perfect Cubes

- 0 cubed is 0³ = 0 × 0 × 0 = 0
- 1 cubed is 1³ = 1 × 1 × 1 = 1
- 2 cubed is 2³ = 2 × 2 × 2 = 8
- 3 cubed is 3³ = 3 × 3 × 3 = 27
- 4 cubed is 4³ = 4 × 4 × 4 = 64
- 5 cubed is 5³ = 5 × 5 × 5 = 125
- 6 cubed is 6³ = 6 × 6 × 6 = 216
- 7 cubed is 7³ = 7 × 7 × 7 = 343
- 8 cubed is 8³ = 8 × 8 × 8 = 512
- 9 cubed is 9³ = 9 × 9 × 9 = 729
- 10 cubed is 10³ = 10 × 10 × 10 = 1000

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## Examples Of Absolute Difference Formula Calculations:

**1. Find the absolute difference between 3 and 9.**

The difference between 3 and 9 on a number line is 6 units.

**2. Find the absolute difference between -3 and 9.**

The difference between -3 and 9 on a number line is 12 units.

**3. Find the absolute difference between -3 and -9.**

The difference between -3 and -9 on a number line is 6 units.

Furey, Edward “Absolute Difference Calculator” at from CalculatorSoup, – Online Calculators

## How To Solve The Logarithmic Equation

If we have the equation used in the Logarithm Equation Calculator

We can say the following is also true

Using the logarithmic function where

We can rewrite our equation to solve for x

Solving for b in equation we have

Solving for y in equation

take the log of both sides:

Using logarithmic identity we rewrite the equation:

Dividing both sides by log b:

Note that writing log without the subscript for the base it is assumed to be log base 10 as in log10.

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## Numbers 0 Through 10 Squared

- 0 squared is 0² = 0 × 0 = 0
- 1 squared is 1² = 1 × 1 = 1
- 2 squared is 2² = 2 × 2 = 4
- 3 squared is 3² = 3 × 3 = 9
- 4 squared is 4² = 4 × 4 = 16
- 5 squared is 5² = 5 × 5 = 25
- 6 squared is 6² = 6 × 6 = 36
- 7 squared is 7² = 7 × 7 = 49
- 8 squared is 8² = 8 × 8 = 64
- 9 squared is 9² = 9 × 9 = 81
- 10 squared is 10² = 10 × 10 = 100

## Example : Solve For Y In The Following Logarithmic Equation

If we have

then it is also true that

Using the logarithmic function we can rewrite the left side of the equation and we get

To solve for y, first take the log of both sides:

Dividing both sides by log 3:

Using a calculator we can find that log 5 0.69897 and log 3 0.4771 2 then our equation becomes:

Therefore, putting y back into our original equation

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## Examples Using The Quadratic Formula

**Example 1:** Find the Solution for \, where a = 1, b = -8 and c = 5, using the Quadratic Formula.

The discriminant \ so, there are two real roots.

Simplify fractions and/or signs:

which becomes

**Example 2:** Find the Solution for \, where a = 5, b = 20 and c = 32, using the Quadratic Formula.

The discriminant \ so, there are two complex roots.

Simplify fractions and/or signs:

which becomes

## Mixed Numbers Calculator :

This online calculator handles simple operations on whole numbers, integers, mixed numbers, fractions and improper fractions by adding, subtracting, dividing or multiplying. The answer is provided in a reduced fraction and a mixed number if it exists.

Enter mixed numbers, whole numbers or fractions in the following formats:

- Mixed numbers: Enter as 1 1/2 which is one and one half or 25 3/32 which is twenty five and three thirty seconds. Keep exactly one space between the whole number and fraction and use a forward slash to input fractions. You can enter up to 3 digits in length for each whole number, numerator or denominator .
- Whole numbers: Up to 3 digits in length.
- Fractions: Enter as 3/4 which is three fourths or 3/100 which is three one hundredths. You can enter up to 3 digits in length for each the numerators and denominators .

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## Order Of Operations Acronyms

The acronyms for order of operations mean you should solve equations in this order always working left to right in your equation.

**PEMDAS** stands for “**P**arentheses, **E**xponents,**M**ultiplication *and***D**ivision, **A**ddition*and***S**ubtraction”

You may also see BEDMAS, BODMAS, and GEMDAS as order of operations acronyms. In these acronyms, “brackets” are the same as parentheses, and “order” is the same as exponents. For GEMDAS, “grouping” is like parentheses or brackets.

**BEDMAS **stands for “**B**rackets, **E**xponents,**D**ivision *and***M**ultiplication, **A**ddition*and***S**ubtraction”

BEDMAS is similar to BODMAS.

**BODMAS **stands for “**B**rackets, **O**rder,**D**ivision *and***M**ultiplication, **A**ddition*and***S**ubtraction”

**GEMDAS **stands for “**G**rouping, **E**xponents,**D**ivision *and***M**ultiplication, **A**ddition*and***S**ubtraction”

**MDAS ** is a subset of the acronyms above. It stands for “**M**ultiplication, *and***D**ivision, **A**ddition *and***S**ubtraction”

## Completing The Square When A Is Not 1

To complete the square when a is greater than 1 or less than 1 but not equal to 0, factor out the value of a from all other terms.

For example, find the solution by completing the square for:

\ so divide through by 2

which gives us

Now, continue to solve this quadratic equation by completing the square method.

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## How To Add Or Subtract Fractions

## Difference Of Two Squares When A Is Negative

If both terms a and b are negative such that we have -a2 – b2 the equation is not in the form of a2 – b2and cannot be rearranged into this form.

If a is negative and we have addition such that we have -a2 + b2 the equation can be rearranged to the form of b2 – a2which is the correct equation only the letters a and b are switched we can just rename our terms.

For example, factor the equation

We can rearrange this equation to

and now solve the difference of two squares with a = 36 and b = 4y2

Solution:

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## Note: Doing Math With Significant Figures

There are some cases where you would*not* want to auto-calculate significant figures. If your calculation involves a constant or an exact value as you might find in a formula, do not check the “auto-calculate” box.

For example, consider the formula for diameter of a circle, d = 2r, where diameter is twice the length of the radius. If you measure a radius of 2.35, multiply by 2 to find the diameter of the circle: 2 * 2.35 = 4.70

If you use this calculator for the calculation and you mark the “auto-calculate” box, the calculator will read the 2 as one significant figure. Your resulting calculation will be rounded from 4.70 to 5, which is clearly not the correct answer to the diameter calculation d=2r.

You can think of constants or exact values as having infinitely many significant figures, or at least as many significant figures as the least precise number in your calculation. Use the appropriate number of significant figures when you input exact values in this calculator. In this example you would want to enter 2.00 for the constant value so that it has the same number of significant figures as the radius entry. The resulting answer would be 4.70 which has 3 significant figures.

## Math Order Of Operations

PEMDAS is an acronym that may help you remember order of operations for solving math equations. PEMDAS is typcially expanded into the phrase, “Please Excuse My Dear Aunt Sally.” The first letter of each word in the phrase creates the PEMDAS acronym. Solve math problems with the standard mathematical order of operations, working left to right:

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## Convert Ratio To Fraction

A part-to-part ratio states the proportion of the parts in relation to each other. The sum of the parts makes up the whole. The ratio 1 : 2 is read as “1 to 2.” This means of the whole of 3, there is a part worth 1 and another part worth 2.

To convert a part-to-part ratio to fractions:

## What Is A Factorial

A factorial is a function that multiplies a number by every number below it. For example 5!= 5*4*3*2*1=120. The function is used, among other things, to find the number of way n objects can be arranged.

- Factorial
- There are n! ways of arranging n distinct objects into an ordered sequence.
- n
- the set or population

In mathematics, there are n! ways to arrange n objects in sequence. “The factorial n! gives the number of ways in which n objects can be permuted.” For example:

- 2 factorial is 2! = 2 x 1 = 2 — There are 2 different ways to arrange the numbers 1 through 2. and .
- 4 factorial is 4! = 4 x 3 x 2 x 1 = 24 — There are 24 different ways to arrange the numbers 1 through 4. , , , , , etc.
- 5 factorial is 5! = 5 x 4 x 3 x 2 x 1 = 120
- 0 factorial is a definition: 0! = 1. There is exactly 1 way to arrange 0 objects.

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## Handshake Problem As A Combinations Problem

We can also solve this Handshake Problem as a combinations problem as C.

n = number of people in the group r = 2, the number of people involved in each different handshake

The order of the items chosen in the subset does not matter so for a group of 3 it will count 1 with 2, 1 with 3, and 2 with 3 but ignore 2 with 1, 3 with 1, and 3 with 2 because these last 3 are duplicates of the first 3 respectively.

expanding the factorials,

cancelling and simplifying,

which is the same as the equation above.