Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
610-(600*(x)+400000/x)=0
Step by step solution :
Step 1 :
400000 Simplify —————— x Equation at the end of step 1 :
400000 610 - (600x + ——————) = 0 x Step 2 :
Rewriting the whole as an Equivalent Fraction :
2.1Adding a fraction to a whole
Rewrite the whole as a fraction using x as the denominator :
600x 600x • x 600x = ———— = ———————— 1 x 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:
600x • x + 400000 600x2 + 400000 ————————————————— = —————————————— x x Equation at the end of step 2 :
(600x2 + 400000) 610 - ———————————————— = 0 x Step 3 :
Rewriting the whole as an Equivalent Fraction :
3.1Subtracting a fraction from a whole
Rewrite the whole as a fraction using x as the denominator :
610 610 • x 610 = ——— = ——————— 1 x Step 4 :
Pulling out like terms :
4.1 Pull out like factors:
600x2 + 400000=200•(3x2 + 2000)
Polynomial Roots Calculator :
4.2 Find roots (zeroes) of : F(x) = 3x2 + 2000
Polynomial Roots Calculator is a set of methods aimed at finding values ofxfor which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational numberP/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 3 and the Trailing Constant is 2000. The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,2 ,4 ,5 ,8 ,10 ,16 ,20 ,25 ,40 , etc Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 2003.00 | ||||||
| -1 | 3 | -0.33 | 2000.33 | ||||||
| -2 | 1 | -2.00 | 2012.00 | ||||||
| -2 | 3 | -0.67 | 2001.33 | ||||||
| -4 | 1 | -4.00 | 2048.00 |
Note - For tidiness, printing of 35 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
4.3 Adding up the two equivalent fractions
610 • x - (200 • (3x2+2000)) -600x2 + 610x - 400000 ———————————————————————————— = —————————————————————— x x Step 5 :
Pulling out like terms :
5.1 Pull out like factors:
-600x2 + 610x - 400000=-10•(60x2 - 61x + 40000)
Trying to factor by splitting the middle term
5.2Factoring 60x2 - 61x + 40000
The first term is, 60x2 its coefficient is 60.
The middle term is, -61x its coefficient is -61.
The last term, "the constant", is +40000
Step-1 : Multiply the coefficient of the first term by the constant 60•40000=2400000
Step-2 : Find two factors of 2400000 whose sum equals the coefficient of the middle term, which is -61.
Numbers too big. Method shall not be applied
Equation at the end of step 5 :
-10 • (60x2 - 61x + 40000) —————————————————————————— = 0 x 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:
-10•(60x2-61x+40000) ———————————————————— • x = 0 • x x Now, on the left hand side, the x cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape:
-10 • (60x2-61x+40000)=0
Equations which are never true:
6.2Solve:-10=0
This equation has no solution.
A a non-zero constant never equals zero.
Parabola, Finding the Vertex:
6.3Find the Vertex ofy = 60x2-61x+40000Parabolas 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,60, 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 0.5083Plugging into the parabola formula 0.5083 for x we can calculate the y-coordinate:
y = 60.0 * 0.51 * 0.51 - 61.0 * 0.51 + 40000.0
or y = 39984.496
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = 60x2-61x+40000
Axis of Symmetry (dashed) {x}={ 0.51}
Vertex at {x,y} = { 0.51,39984.50}
Function has no real roots
Solve Quadratic Equation using the Quadratic Formula
6.4Solving60x2-61x+40000 = 0 by the Quadratic Formula.According to the Quadratic Formula,x, the solution forAx2+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= 60.00
B= -61.00
C= 40000.00 B2 = 3721.00
4AC = 9600000.00
B2-4AC = -9596279.00
SQRT(B2-4AC)= 3097.79 i
x=( 61.00± 3097.79 i )/ 120.00
x= 0.50833- 25.81488i
x= 0.50833- 25.81488i
Two solutions were found :
- x= 0.50833- 25.81488i
- x= 0.50833- 25.81488i
