##### 3/2 5/x-3=x 9/2x-6

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Algebra

Jul 16th, 2015

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## Two solutions were found :

1.  x =(-9-√521)/-22= 1.447
2.  x =(-9+√521)/-22=-0.628

### Rearrange:

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

3/2+5/x-3-(x+9/2*x-6)=0

## Step  1  :

```                 9x
Simplify   x  +  ——
2
```

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

1.1  Adding a fraction to a whole

Rewrite the whole as a fraction using   as the denominator :

```          x     x • 2
x =  —  =  —————
1       2
```

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 :

1.2   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:

``` x • 2 + 9x     11x
——————————  =  ———
2           2
```

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

```    3 5      11x
((—+—)-3)-(———-6)  = 0
2 x       2
```

## Step  2  :

```           11x
Simplify   ———  -  6
2
```

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

2.1  Subtracting a whole from a fraction

Rewrite the whole as a fraction using   as the denominator :

```          6     6 • 2
6 =  —  =  —————
1       2
```

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

2.2   Adding up the two equivalent fractions

``` 11x - (6 • 2)     11x - 12
—————————————  =  ————————
2              2
```

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

```    3    5           (11x - 12)
((— +  —) -  3) -  ——————————  = 0
2    x               2
```

## Step  3  :

```           3     5
Simplify   —  +  —
2     x
```

#### Calculating the Least Common Multiple :

3.1   Find the Least Common Multiple

The left denominator is :

The right denominator is :

Number of times each prime factor
appears in the factorization of:
Prime
Factor
Left
Denominator
Right
Denominator
L.C.M = Max
{Left,Right}
2101
Product of all
Prime Factors
212
Number of times each Algebraic Factor
appears in the factorization of:
Algebraic
Factor
Left
Denominator
Right
Denominator
L.C.M = Max
{Left,Right}
x 011

Least Common Multiple:
2x

#### Calculating Multipliers :

3.2   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 = x

Right_M = L.C.M / R_Deno = 2

#### Making Equivalent Fractions :

3.3   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  (y2+y)/(y+1)3 are equivalent as well.

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

```   L. Mult. • L. Num.      3 • x
——————————————————  =   —————
L.C.M              2x

R. Mult. • R. Num.      5 • 2
——————————————————  =   —————
L.C.M              2x
```

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

3.4   Adding up the two equivalent fractions

``` 3 • x + 5 • 2     3x + 10
—————————————  =  ———————
2x             2x
```

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

```   (3x + 10)          (11x - 12)
(————————— -  3) -  ——————————  = 0
2x                  2
```

## Step  4  :

```           3x+10
Simplify   —————  -  3
2x
```

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

4.1  Subtracting a whole from a fraction

Rewrite the whole as a fraction using  2x  as the denominator :

```          3     3 • 2x
3 =  —  =  ——————
1       2x
```

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

4.2   Adding up the two equivalent fractions

``` (3x+10) - (3 • 2x)     10 - 3x
——————————————————  =  ———————
2x               2x
```

#### Equation at the end of step  4  :

```  (10 - 3x)    (11x - 12)
————————— -  ——————————  = 0
2x            2
```

## Step  5  :

```           10-3x     11x-12
Simplify   —————  -  ——————
2x         2
```

#### Calculating the Least Common Multiple :

5.1   Find the Least Common Multiple

The left denominator is :   2x

The right denominator is :

Number of times each prime factor
appears in the factorization of:
Prime
Factor
Left
Denominator
Right
Denominator
L.C.M = Max
{Left,Right}
2111
Product of all
Prime Factors
222
Number of times each Algebraic Factor
appears in the factorization of:
Algebraic
Factor
Left
Denominator
Right
Denominator
L.C.M = Max
{Left,Right}
x 101

Least Common Multiple:
2x

#### Calculating Multipliers :

5.2   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 = 1

Right_M = L.C.M / R_Deno = x

#### Making Equivalent Fractions :

5.3   Rewrite the two fractions into equivalent fractions

```   L. Mult. • L. Num.      (10-3x)
——————————————————  =   ———————
L.C.M               2x

R. Mult. • R. Num.      (11x-12) • x
——————————————————  =   ————————————
L.C.M                  2x
```

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

5.4   Adding up the two equivalent fractions

``` (10-3x) - ((11x-12) • x)     -11x2 + 9x + 10
————————————————————————  =  ———————————————
2x                      2x
```

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

5.5  Factoring  -11x2 + 9x + 10

The first term is,  -11x2  its coefficient is  -11 .
The middle term is,  +9x  its coefficient is  .
The last term, "the constant", is  +10

Step-1 : Multiply the coefficient of the first term by the constant   -11 • 10 = -110

Step-2 : Find two factors of  -110  whose sum equals the coefficient of the middle term, which is   .

 -110 + 1 = -109 -55 + 2 = -53 -22 + 5 = -17 -11 + 10 = -1 -10 + 11 = 1 -5 + 22 = 17 -2 + 55 = 53 -1 + 110 = 109

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

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

```  -11x2 + 9x + 10
———————————————  = 0
2x
```

## Step  6  :

```        -11x2+9x+10
Solve   ———————————  = 0
2x
```

#### 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:

```  -11x2+9x+10
——————————— • 2x = 0 • 2x
2x
```

Now, on the left hand side, the  2x  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
-11x2+9x+10 = 0

#### Parabola, Finding the Vertex :

6.2  Find the Vertex of y = -11x2+9x+10Parabolas 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, -11 , 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,Ax2+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   0.4091
Plugging into the parabola formula   0.4091  for  x  we can calculate the  y -coordinate :  y = -11.0 * 0.41 * 0.41 + 9.0 * 0.41 + 10.0 or  y = 11.841

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

``` Root plot for :  y = -11x2+9x+10
Axis of Symmetry (dashed)  {x}={ 0.41}
Vertex at  {x,y} = { 0.41,11.84}
x -Intercepts (Roots) :
Root 1  at  {x,y} = { 1.45, 0.00}
Root 2  at  {x,y} = {-0.63, 0.00}

```

#### Solve Quadratic Equation by Completing The Square

6.3  Solving  -11x2+9x+10 = 0 by Completing The Square .
Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
11x2-9x-10 = 0 Divide both sides of the equation by  11  to have 1 as the coefficient of the first term :
x2-(9/11)x-(10/11) = 0

Add  10/11  to both side of the equation :
x2-(9/11)x = 10/11

Now the clever bit: Take the coefficient of  x , which is  9/11 , divide by two, giving  9/22 , and finally square it giving  81/484

Add  81/484  to both sides of the equation :
On the right hand side we have :
10/11 +  81/484 The common denominator of the two fractions is  484  Adding (440/484)+(81/484)  gives  521/484
So adding to both sides we finally get :
x2-(9/11)x+(81/484) = 521/484

Adding  81/484  has completed the left hand side into a perfect square :
x2-(9/11)x+(81/484)  =
(x-(9/22)) • (x-(9/22)) =
(x-(9/22))2
Things which are equal to the same thing are also equal to one another. Since
x2-(9/11)x+(81/484) = 521/484 and
x2-(9/11)x+(81/484) = (x-(9/22))2
then, according to the law of transitivity,
(x-(9/22))2 = 521/484

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
(x-(9/22))2   is
(x-(9/22))2/2 =
(x-(9/22))1 =
x-(9/22)

Now, applying the Square Root Principle to  Eq. #6.3.1  we get:
x-(9/22) = √ 521/484

Add  9/22  to both sides to obtain:
x = 9/22 + √ 521/484

Since a square root has two values, one positive and the other negative
x2 - (9/11)x - (10/11) = 0
has two solutions:
x = 9/22 + √ 521/484
or
x = 9/22 - √ 521/484

Note that  √ 521/484 can be written as
√ 521  / √ 484  which is  521  / 22

### Solve Quadratic Equation using the Quadratic Formula

6.4  Solving  -11x2+9x+10 = 0 by the Quadratic Formula .
According to the Quadratic Formula, x , the solution for  Ax2+Bx+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :

- B ±  √ B2-4AC
x =  ————————
2A  In our case, A = -11
B =  9
C =  10 Accordingly, B2 - 4AC =
81 - (-440) =
521Applying the quadratic formula :

-9 ± √ 521
x = ——————
-22  √ 521   , rounded to 4 decimal digits, is  22.8254 So now we are looking at:
x = ( -9 ±  22.825 ) / -22

Two real solutions:

x =(-9+√521)/-22=-0.628

or:

x =(-9-√521)/-22= 1.447

Please let me know if you need any clarification. I'm always happy to answer your questions.
Jul 16th, 2015

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Jul 16th, 2015
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Jul 16th, 2015
May 22nd, 2017
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