Step by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "0.4" was replaced by "(4/10)".
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
0-(120*t-(4/10)*t^4+1000)=0
Step by step solution :
Step 1 :
2
Simplify —
5
Equation at the end of step 1 :
2
0 - ((120t - (— • t4)) + 1000) = 0
5
Step 2 :
Equation at the end of step 2 :
2t4
0 - ((120t - ———) + 1000) = 0
5
Step 3 :
Rewriting the whole as an Equivalent Fraction :
3.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using 5 as the denominator :
120t 120t • 5
120t = ———— = ————————
1 5
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:
120t • 5 - (2t4) 600t - 2t4
———————————————— = ——————————
5 5
Equation at the end of step 3 :
(600t - 2t4)
0 - (———————————— + 1000) = 0
5
Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Adding a whole to a fraction
Rewrite the whole as a fraction using 5 as the denominator :
1000 1000 • 5
1000 = ———— = ————————
1 5
Step 5 :
Pulling out like terms :
5.1 Pull out like factors :
600t - 2t4 = -2t • (t3 - 300)
Trying to factor as a Difference of Cubes:
5.2 Factoring: t3 - 300
Theory : A difference of two perfect cubes, a3 - b3 can be factored into
(a-b) • (a2 +ab +b2)
Proof : (a-b)•(a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3+(a2b-ba2)+(ab2-b2a)-b3 =
a3+0+0-b3 =
a3-b3
Check : 300 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Polynomial Roots Calculator :
5.3 Find roots (zeroes) of : F(t) = t3 - 300
Polynomial Roots Calculator is a set of methods aimed at finding values of t for which F(t)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers t which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/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 1 and the Trailing Constant is -300.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,5 ,6 ,10 ,12 ,15 ,20 , etc
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -301.00 | ||||||
| -2 | 1 | -2.00 | -308.00 | ||||||
| -3 | 1 | -3.00 | -327.00 | ||||||
| -4 | 1 | -4.00 | -364.00 | ||||||
| -5 | 1 | -5.00 | -425.00 |
Note - For tidiness, printing of 15 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
5.4 Adding up the two equivalent fractions
-2t • (t3-300) + 1000 • 5 -2t4 + 600t + 5000
————————————————————————— = ——————————————————
5 5
Equation at the end of step 5 :
(-2t4 + 600t + 5000)
0 - ———————————————————— = 0
5
Step 6 :
Step 7 :
Pulling out like terms :
7.1 Pull out like factors :
2t4 - 600t - 5000 = 2 • (t4 - 300t - 2500)
Polynomial Roots Calculator :
7.2 Find roots (zeroes) of : F(t) = t4 - 300t - 2500
See theory in step 5.3
In this case, the Leading Coefficient is 1 and the Trailing Constant is -2500.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4 ,5 ,10 ,20 ,25 ,50 ,100 ,125 , etc
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -2199.00 | ||||||
| -2 | 1 | -2.00 | -1884.00 | ||||||
| -4 | 1 | -4.00 | -1044.00 | ||||||
| -5 | 1 | -5.00 | -375.00 | ||||||
| -10 | 1 | -10.00 | 10500.00 |
Note - For tidiness, printing of 15 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Equation at the end of step 7 :
2 • (t4 - 300t - 2500)
—————————————————————— = 0
5
Step 8 :
When a fraction equals zero :
8.1 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:
2•(t4-300t-2500)
———————————————— • 5 = 0 • 5
5
Now, on the left hand side, the 5 cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape :
2 • (t4-300t-2500) = 0
Equations which are never true :
8.2 Solve : 2 = 0
This equation has no solution.
A a non-zero constant never equals zero.
Quartic Equations :
8.3 Solve t4-300t-2500 = 0
In search of an interavl at which the above polynomial changes sign, from negative to positive or the other wayaround.
Method of search: Calculate polynomial values for all integer points between t=-20 and t=+20
Found change of sign between t= -6.00 and t= -5.00
Approximating a root using the Bisection Method :
We now use the Bisection Method to approximate one of the solutions. The Bisection Method is an iterative procedure to approximate a root (Root is another name for a solution of an equation).
The function is F(t) = t4 - 300t - 2500
At t= -5.00 F(t) is equal to -375.00
At t= -6.00 F(t) is equal to 596.00
Intuitively we feel, and justly so, that since F(t) is negative on one side of the interval, and positive on the other side then, somewhere inside this interval, F(t) is zero
Procedure :
(1) Find a point "Left" where F(Left) < 0
(2) Find a point 'Right' where F(Right) > 0
(3) Compute 'Middle' the middle point of the interval [Left,Right]
(4) Calculate Value = F(Middle)
(5) If Value is close enough to zero goto Step (7)
Else :
If Value < 0 then : Left <- Middle
If Value > 0 then : Right <- Middle
(6) Loop back to Step (3)
(7) Done!! The approximation found is Middle
Follow Middle movements to understand how it works :
Left Value(Left) Right Value(Right) -5.000000000 -375.000000000 -6.000000000 596.000000000 0.000000000 -2500.000 -6.000000000 596.000000000 -3.000000000 -1519.000 -6.000000000 596.000000000 -4.500000000 -739.937500000 -6.000000000 596.000000000 -5.250000000 -165.308593750 -6.000000000 596.000000000 -5.250000000 -165.308593750 -5.625000000 188.629150391 -5.250000000 -165.308593750 -5.437500000 5.422378540 -5.343750000 -81.449049950 -5.437500000 5.422378540 -5.390625000 -38.396440446 -5.437500000 5.422378540 -5.414062500 -16.583640870 -5.437500000 5.422378540 -5.425781250 -5.604888256 -5.437500000 5.422378540 -5.431640625 -0.097332232 -5.437500000 5.422378540 -5.431640625 -0.097332232 -5.434570312 2.661002171 -5.431640625 -0.097332232 -5.433105469 1.281454929 -5.431640625 -0.097332232 -5.432373047 0.591966364 -5.431640625 -0.097332232 -5.432006836 0.247293323 -5.431640625 -0.097332232 -5.431823730 0.074974610 -5.431732178 -0.011180295 -5.431823730 0.074974610 -5.431732178 -0.011180295 -5.431777954 0.031896786 -5.431732178 -0.011180295 -5.431755066 0.010358153 -5.431743622 -0.000411094 -5.431755066 0.010358153 -5.431743622 -0.000411094 -5.431749344 0.004973524 -5.431743622 -0.000411094 -5.431746483 0.002281213 -5.431743622 -0.000411094 -5.431745052 0.000935059 -5.431743622 -0.000411094 -5.431744337 0.000261982 -5.431743979 -0.000074556 -5.431744337 0.000261982 -5.431743979 -0.000074556 -5.431744158 0.000093713 -5.431743979 -0.000074556 -5.431744069 0.000009579 -5.431744024 -0.000032489 -5.431744069 0.000009579
Next Middle will get us close enough to zero:
F( -5.431744058 ) is -0.000000938
The desired approximation of the solution is:
t ≓ -5.431744058
Note, ≓ is the approximation symbol
One solution was found :
t ≓ -5.431744058How did we do?
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