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): "0.064" was replaced by "(064/1000)". 3 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 :
(25/1000)-(((112/1000)-x)*((64/1000)-x)/x)=0
Step by step solution :
Step 1 :
8
Simplify ———
125
Equation at the end of step 1 :
25 112 (8
————-((————-x)•————) = 0
1000 1000 125x
Step 2 :
Rewriting the whole as an Equivalent Fraction :
2.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using 125 as the denominator :
x x • 125
x = — = ———————
1 125
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:
8 - (x • 125) 8 - 125x
————————————— = ————————
125 125
Equation at the end of step 2 :
25 112 (8-125x)
————-((————-x)•———————— ÷ x) = 0
1000 1000 125
Step 3 :
8-125x
Divide —————— by x
125
Equation at the end of step 3 :
25 112 (8 - 125x)
———— - ((———— - x) • ——————————) = 0
1000 1000 125x
Step 4 :
14
Simplify ———
125
Equation at the end of step 4 :
25 14 (8 - 125x)
———— - ((——— - x) • ——————————) = 0
1000 125 125x
Step 5 :
Rewriting the whole as an Equivalent Fraction :
5.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using 125 as the denominator :
x x • 125
x = — = ———————
1 125
Adding fractions that have a common denominator :
5.2 Adding up the two equivalent fractions
14 - (x • 125) 14 - 125x
—————————————— = —————————
125 125
Equation at the end of step 5 :
25 (14 - 125x) (8 - 125x)
———— - (——————————— • ——————————) = 0
1000 125 125x
Step 6 :
Equation at the end of step 6 :
25 (14 - 125x) • (8 - 125x)
———— - ———————————————————————— = 0
1000 15625x
Step 7 :
1
Simplify ——
40
Equation at the end of step 7 :
1 (14 - 125x) • (8 - 125x)
—— - ———————————————————————— = 0
40 15625x
Step 8 :
Calculating the Least Common Multiple :
8.1 Find the Least Common Multiple
The left denominator is : 40
The right denominator is : 15625x
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 3 | 0 | 3 |
| 5 | 1 | 6 | 6 |
| Product of all Prime Factors | 40 | 15625 | 125000 |
| Algebraic Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| x | 0 | 1 | 1 |
Least Common Multiple:
125000x
Calculating Multipliers :
8.2 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 = 3125x
Right_M = L.C.M / R_Deno = 8
Making Equivalent Fractions :
8.3 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. 3125x —————————————————— = ——————— L.C.M 125000x R. Mult. • R. Num. (14-125x) • (8-125x) • 8 —————————————————— = ———————————————————————— L.C.M 125000x
Adding fractions that have a common denominator :
8.4 Adding up the two equivalent fractions
3125x - ((14-125x) • (8-125x) • 8) -125000x2 + 25125x - 896
—————————————————————————————————— = ————————————————————————
125000x 125000x
Step 9 :
Pulling out like terms :
9.1 Pull out like factors :
-125000x2 + 25125x - 896 = -1 • (125000x2 - 25125x + 896)
Trying to factor by splitting the middle term
9.2 Factoring 125000x2 - 25125x + 896
The first term is, 125000x2 its coefficient is 125000 .
The middle term is, -25125x its coefficient is -25125 .
The last term, "the constant", is +896
Step-1 : Multiply the coefficient of the first term by the constant 125000 • 896 = 112000000
Step-2 : Find two factors of 112000000 whose sum equals the coefficient of the middle term, which is -25125 .
Numbers too big. Method shall not be applied
Equation at the end of step 9 :
-125000x2 + 25125x - 896
———————————————————————— = 0
125000x
Step 10 :
When a fraction equals zero :
10.1 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:
-125000x2+25125x-896
———————————————————— • 125000x = 0 • 125000x
125000x
Now, on the left hand side, the 125000x cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape :
-125000x2+25125x-896 = 0
Parabola, Finding the Vertex :
10.2 Find the Vertex of y = -125000x2+25125x-896
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, -125000 , 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 0.1005
Plugging into the parabola formula 0.1005 for x we can calculate the y -coordinate :
y = -125000.0 * 0.10 * 0.10 + 25125.0 * 0.10 - 896.0
or y = 366.531
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = -125000x2+25125x-896
Axis of Symmetry (dashed) {x}={ 0.10}
Vertex at {x,y} = { 0.10,366.53}
x -Intercepts (Roots) :
Root 1 at {x,y} = { 0.15, 0.00}
Root 2 at {x,y} = { 0.05, 0.00}
Solve Quadratic Equation using the Quadratic Formula
10.3 Solving -125000x2+25125x-896 = 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 = -125000.00
B = 25125.00
C = -896.00
B2 = 631265625.00
4AC = 448000000.00
B2 - 4AC = 183265625.00
SQRT(B2-4AC) = 13537.56
x=( -25125.00 ± 13537.56) / -250000.00
x = 0.04635
x = 0.15465
Two solutions were found :
- x = 0.15465
- x = 0.04635
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