Solution - Adding, subtracting and finding the least common multiple
Other Ways to Solve
Adding, subtracting and finding the least common multipleStep by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "0.0129" was replaced by "(0129/10000)". 4 more similar replacement(s)
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
(((52/1000)-x)*((898/10000)-x))/((21/100)+x)-((129/10000))=0
Step by step solution :
Step 1 :
129
Simplify —————
10000
Equation at the end of step 1 :
(52 898 21 129
————-x)•(—————-x))———+x)-————— = 0
1000 10000( 100 10000
Step 2 :
21
Simplify ———
100
Equation at the end of step 2 :
(52 898 21 129
————-x)•(—————-x))———+x)-————— = 0
1000 10000( 100 10000
Step 3 :
Rewriting the whole as an Equivalent Fraction :
3.1 Adding a whole to a fraction
Rewrite the whole as a fraction using 100 as the denominator :
x x • 100
x = — = ———————
1 100
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 :
3.2 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:
21 + x • 100 100x + 21
———————————— = —————————
100 100
Equation at the end of step 3 :
(52 898 (100x+21) 129
————-x)•(—————-x))—————————-————— = 0
1000 10000 100 10000
Step 4 :
449
Simplify ————
5000
Equation at the end of step 4 :
(52 449 (100x+21) 129
————-x)•(————-x))—————————-————— = 0
1000 5000 100 10000
Step 5 :
Rewriting the whole as an Equivalent Fraction :
5.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using 5000 as the denominator :
x x • 5000
x = — = ————————
1 5000
Adding fractions that have a common denominator :
5.2 Adding up the two equivalent fractions
449 - (x • 5000) 449 - 5000x
———————————————— = ———————————
5000 5000
Equation at the end of step 5 :
(52 (449-5000x) (100x+21) 129
————-x)•———————————) ÷ —————————-————— = 0
1000 5000 100 10000
Step 6 :
13
Simplify ———
250
Equation at the end of step 6 :
(13 (449 - 5000x) (100x + 21) 129
——— - x) • —————————————) ÷ ——————————— - ————— = 0
250 5000 100 10000
Step 7 :
Rewriting the whole as an Equivalent Fraction :
7.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using 250 as the denominator :
x x • 250
x = — = ———————
1 250
Adding fractions that have a common denominator :
7.2 Adding up the two equivalent fractions
13 - (x • 250) 13 - 250x
—————————————— = —————————
250 250
Equation at the end of step 7 :
(13 - 250x) (449 - 5000x) (100x + 21) 129
——————————— • —————————————) ÷ ——————————— - ————— = 0
250 5000 100 10000
Step 8 :
Equation at the end of step 8 :
(13 - 250x) • (449 - 5000x) (100x + 21) 129
——————————————————————————— ÷ ——————————— - ————— = 0
1250000 100 10000
Step 9 :
(13-250x)•(449-5000x) 100x+21
Divide ————————————————————— by ———————
1250000 100
9.1 Dividing fractions
To divide fractions, write the divison as multiplication by the reciprocal of the divisor :
(13 - 250x) • (449 - 5000x) 100x + 21 (13 - 250x) • (449 - 5000x) 100 ——————————————————————————— ÷ ————————— = ——————————————————————————— • ——————————— 1250000 100 1250000 (100x + 21)
Equation at the end of step 9 :
100 • (13 - 250x) • (449 - 5000x) 129
————————————————————————————————— - ————— = 0
1250000 • (100x + 21) 10000
Step 10 :
Calculating the Least Common Multiple :
10.1 Find the Least Common Multiple
The left denominator is : 1250000 • (100x+21)
The right denominator is : 10000
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 4 | 4 | 4 |
| 5 | 7 | 4 | 7 |
| Product of all Prime Factors | 1250000 | 10000 | 1250000 |
| Algebraic Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 100x+21 | 1 | 0 | 1 |
Least Common Multiple:
1250000 • (100x+21)
Calculating Multipliers :
10.2 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 = 125•(100x+21)
Making Equivalent Fractions :
10.3 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 respective Multiplier.
L. Mult. • L. Num. 100 • (13-250x) • (449-5000x) —————————————————— = ————————————————————————————— L.C.M 1250000 • (100x+21) R. Mult. • R. Num. 129 • 125 • (100x+21) —————————————————— = ————————————————————— L.C.M 1250000 • (100x+21)
Adding fractions that have a common denominator :
10.4 Adding up the two equivalent fractions
100 • (13-250x) • (449-5000x) - (129 • 125 • (100x+21)) 125000000x2-19337500x+245075
——————————————————————————————————————————————————————— = ————————————————————————————
1250000 • (100x+21) 1250000 • (100x+21)
Step 11 :
Pulling out like terms :
11.1 Pull out like factors :
125000000x2 - 19337500x + 245075 = 25 • (5000000x2 - 773500x + 9803)
Trying to factor by splitting the middle term
11.2 Factoring 5000000x2 - 773500x + 9803
The first term is, 5000000x2 its coefficient is 5000000 .
The middle term is, -773500x its coefficient is -773500 .
The last term, "the constant", is +9803
Step-1 : Multiply the coefficient of the first term by the constant
Numbers too big. Method shall not be applied
Equation at the end of step 11 :
25 • (5000000x2 - 773500x + 9803)
————————————————————————————————— = 0
1250000 • (100x + 21)
Step 12 :
When a fraction equals zero :
12.1 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:
25•(5000000x2-773500x+9803)
——————————————————————————— • 1250000•(100x+21) = 0 • 1250000•(100x+21)
1250000•(100x+21)
Now, on the left hand side, the 1250000 • 100x+21 cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape :
25 • (5000000x2-773500x+9803) = 0
Equations which are never true :
12.2 Solve : 25 = 0
This equation has no solution.
A a non-zero constant never equals zero.
Parabola, Finding the Vertex :
12.3 Find the Vertex of y = 5000000x2-773500x+9803
Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 5000000 , is positive (greater 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 0.0774
Plugging into the parabola formula 0.0774 for x we can calculate the y -coordinate :
y = 5000000.0 * 0.08 * 0.08 - 773500.0 * 0.08 + 9803.0
or y = -20112.112
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = 5000000x2-773500x+9803
Axis of Symmetry (dashed) {x}={ 0.08}
Vertex at {x,y} = { 0.08,-20112.11}
x -Intercepts (Roots) :
Root 1 at {x,y} = { 0.01, 0.00}
Root 2 at {x,y} = { 0.14, 0.00}
Solve Quadratic Equation using the Quadratic Formula
12.4 Solving 5000000x2-773500x+9803 = 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 = 5000000.00
B = -773500.00
C = 9803.00
B2 = 598302250000.00
4AC = 196060000000.00
B2 - 4AC = 402242250000.00
SQRT(B2-4AC) = 634225.71
x=( 773500.00 ± 634225.71) /10000000.00
x = 0.14077
x = 0.01393
Two solutions were found :
- x = 0.01393
- x = 0.14077
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