# Adding, subtracting and finding the least common multiple

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This solution deals with adding, subtracting and finding the least common multiple.

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- Adding, subtracting and finding the least common multiple

## Step by Step Solution

### Reformatting the input :

Changes made to your input should not affect the solution:

(1): "/-4h" was replaced by "/(-4h)".

### Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

21/(-4*h)-(14/11+h)=0

## Step by step solution :

## Step 1 :

```
14
Simplify ——
11
```

#### Equation at the end of step 1 :

```
21 14
——— - (—— + h) = 0
-4h 11
```

## Step 2 :

#### Rewriting the whole as an Equivalent Fraction :

2.1 Adding a whole to a fraction

Rewrite the whole as a fraction using 11 as the denominator :

```
h h • 11
h = — = ——————
1 11
```

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

#### Adding fractions that have a common denominator :

2.2 Adding up the two equivalent fractions

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

```
14 + h • 11 11h + 14
——————————— = ————————
11 11
```

#### Equation at the end of step 2 :

```
21 (11h + 14)
——— - —————————— = 0
-4h 11
```

## Step 3 :

```
21
Simplify ———
-4h
```

#### Equation at the end of step 3 :

```
21 (11h + 14)
——— - —————————— = 0
-4h 11
```

## Step 4 :

#### Calculating the Least Common Multiple :

4.1 Find the Least Common Multiple

The left denominator is : -4h

The right denominator is : 11

Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
---|---|---|---|

2 | 2 | 0 | 2 |

11 | 0 | 1 | 1 |

Product of all Prime Factors | -4 | 11 | 44 |

Algebraic Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
---|---|---|---|

h | 1 | 0 | 1 |

Least Common Multiple:

44h

#### Calculating Multipliers :

4.2 Calculate multipliers for the two fractions

Denote the Least Common Multiple by L.C.M

Denote the Left Multiplier by Left_M

Denote the Right Multiplier by Right_M

Denote the Left Deniminator by L_Deno

Denote the Right Multiplier by R_Deno

Left_M = L.C.M / L_Deno = -11

Right_M = L.C.M / R_Deno = 4h

#### Making Equivalent Fractions :

4.3 Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example : 1/2 and 2/4 are equivalent, y/(y+1)^{2} and (y^{2}+y)/(y+1)^{3} are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

L. Mult. • L. Num. 21 • -11 —————————————————— = ———————— L.C.M 44h R. Mult. • R. Num. (11h+14) • 4h —————————————————— = ————————————— L.C.M 44h

#### Adding fractions that have a common denominator :

4.4 Adding up the two equivalent fractions

` 21 • -11 - ((11h+14) • 4h) -44h`^{2} - 56h - 231
—————————————————————————— = —————————————————
44h 44h

## Step 5 :

#### Pulling out like terms :

5.1 Pull out like factors :

-44h^{2} - 56h - 231 = -1 • (44h^{2} + 56h + 231)

#### Trying to factor by splitting the middle term

5.2 Factoring 44h^{2} + 56h + 231

The first term is, 44h^{2} its coefficient is 44 .

The middle term is, +56h its coefficient is 56 .

The last term, "the constant", is +231

Step-1 : Multiply the coefficient of the first term by the constant 44 • 231 = 10164

Step-2 : Find two factors of 10164 whose sum equals the coefficient of the middle term, which is 56 .

-10164 | + | -1 | = | -10165 | ||

-5082 | + | -2 | = | -5084 | ||

-3388 | + | -3 | = | -3391 | ||

-2541 | + | -4 | = | -2545 | ||

-1694 | + | -6 | = | -1700 | ||

-1452 | + | -7 | = | -1459 |

For tidiness, printing of 66 lines which failed to find two such factors, was suppressed

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

#### Equation at the end of step 5 :

` -44h`^{2} - 56h - 231
————————————————— = 0
44h

## Step 6 :

#### When a fraction equals zero :

` 6.1 When a fraction equals zero ...`

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

` -44h`^{2}-56h-231
————————————— • 44h = 0 • 44h
44h

Now, on the left hand side, the 44h cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :

-44h^{2}-56h-231 = 0

#### Parabola, Finding the Vertex :

6.2 Find the Vertex of y = -44h^{2}-56h-231

Parabolas have a highest or a lowest point called the Vertex . Our parabola opens down and accordingly has a highest point (AKA absolute maximum) . We know this even before plotting "y" because the coefficient of the first term, -44 , is negative (smaller than zero).

Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.

Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.

For any parabola,Ah^{2}+Bh+C,the h -coordinate of the vertex is given by -B/(2A) . In our case the h coordinate is -0.6364

Plugging into the parabola formula -0.6364 for h we can calculate the y -coordinate :

y = -44.0 * -0.64 * -0.64 - 56.0 * -0.64 - 231.0

or y = -213.182

#### Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y = -44h^{2}-56h-231

Axis of Symmetry (dashed) {h}={-0.64}

Vertex at {h,y} = {-0.64,-213.18}

Function has no real roots

#### Solve Quadratic Equation by Completing The Square

6.3 Solving -44h^{2}-56h-231 = 0 by Completing The Square .

Multiply both sides of the equation by (-1) to obtain positive coefficient for the first term:

44h^{2}+56h+231 = 0 Divide both sides of the equation by 44 to have 1 as the coefficient of the first term :

h^{2}+(14/11)h+(21/4) = 0

Subtract 21/4 from both side of the equation :

h^{2}+(14/11)h = -21/4

Now the clever bit: Take the coefficient of h , which is 14/11 , divide by two, giving 7/11 , and finally square it giving 49/121

Add 49/121 to both sides of the equation :

On the right hand side we have :

-21/4 + 49/121 The common denominator of the two fractions is 484 Adding (-2541/484)+(196/484) gives -2345/484

So adding to both sides we finally get :

h^{2}+(14/11)h+(49/121) = -2345/484

Adding 49/121 has completed the left hand side into a perfect square :

h^{2}+(14/11)h+(49/121) =

(h+(7/11)) • (h+(7/11)) =

(h+(7/11))^{2}

Things which are equal to the same thing are also equal to one another. Since

h^{2}+(14/11)h+(49/121) = -2345/484 and

h^{2}+(14/11)h+(49/121) = (h+(7/11))^{2}

then, according to the law of transitivity,

(h+(7/11))^{2} = -2345/484

We'll refer to this Equation as Eq. #6.3.1

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of

(h+(7/11))^{2} is

(h+(7/11))^{2/2} =

(h+(7/11))^{1} =

h+(7/11)

Now, applying the Square Root Principle to Eq. #6.3.1 we get:

h+(7/11) = √ -2345/484

Subtract 7/11 from both sides to obtain:

h = -7/11 + √ -2345/484

In Math, i is called the imaginary unit. It satisfies i^{2} =-1. Both i and -i are the square roots of -1

Since a square root has two values, one positive and the other negative

h^{2} + (14/11)h + (21/4) = 0

has two solutions:

h = -7/11 + √ 2345/484 • i

or

h = -7/11 - √ 2345/484 • i

Note that √ 2345/484 can be written as

√ 2345 / √ 484 which is √ 2345 / 22

### Solve Quadratic Equation using the Quadratic Formula

6.4 Solving -44h^{2}-56h-231 = 0 by the Quadratic Formula .

According to the Quadratic Formula, h , the solution for Ah^{2}+Bh+C = 0 , where A, B and C are numbers, often called coefficients, is given by :

__ __

- B ± √ B^{2}-4AC

h = ————————

2A

In our case, A = -44

B = -56

C = -231

Accordingly, B^{2} - 4AC =

3136 - 40656 =

-37520

Applying the quadratic formula :

56 ± √ -37520

h = ————————

-88

In the set of real numbers, negative numbers do not have square roots. A new set of numbers, called complex, was invented so that negative numbers would have a square root. These numbers are written (a+b*i)

Both i and -i are the square roots of minus 1

Accordingly,√ -37520 =

√ 37520 • (-1) =

√ 37520 • √ -1 =

± √ 37520 • i

Can √ 37520 be simplified ?

Yes! The prime factorization of 37520 is

2•2•2•2•5•7•67

To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).

√ 37520 = √ 2•2•2•2•5•7•67 =2•2•√ 2345 =

± 4 • √ 2345

√ 2345 , rounded to 4 decimal digits, is 48.4252

So now we are looking at:

h = ( 56 ± 4 • 48.425 i ) / -88

Two imaginary solutions :

h =(56+√-37520)/-88=7/-11-i/22√ 2345 = -0.6364+2.2011i or:

h =(56-√-37520)/-88=7/-11+i/22√ 2345 = -0.6364-2.2011i

## Two solutions were found :

- h =(56-√-37520)/-88=7/-11+i/22√ 2345 = -0.6364-2.2011i
- h =(56+√-37520)/-88=7/-11-i/22√ 2345 = -0.6364+2.2011i