found this from https://www.tiger-algebra.com/
Step-by-step explanation:
One solution was found :
x ≓ -0.788228288
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(((4 • (x3)) - 3x2) - 100x) - 75 = 0
Step 2 :
Equation at the end of step 2 :
((22x3 - 3x2) - 100x) - 75 = 0
Step 3 :
Checking for a perfect cube :
3.1 4x3-3x2-100x-75 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: 4x3-3x2-100x-75
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -3x2-75
Group 2: 4x3-100x
Pull out from each group separately :
Group 1: (x2+25) • (-3)
Group 2: (x2-25) • (4x)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Polynomial Roots Calculator :
3.3 Find roots (zeroes) of : F(x) = 4x3-3x2-100x-75
Polynomial Roots Calculator is a set of methods aimed at finding values of x for 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 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 4 and the Trailing Constant is -75.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4
of the Trailing Constant : 1 ,3 ,5 ,15 ,25 ,75
Let us test
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 18.00
-1 2 -0.50 -26.25
-1 4 -0.25 -50.25
-3 1 -3.00 90.00
-3 2 -1.50 54.75
Note - For tidiness, printing of 31 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Equation at the end of step 3 :
4x3 - 3x2 - 100x - 75 = 0
Step 4 :
Cubic Equations :
4.1 Solve 4x3-3x2-100x-75 = 0
Future releases of Tiger-Algebra will solve equations of the third degree directly.
Meanwhile we will use the Bisection method to approximate one real solution.
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(x) = 4x3 - 3x2 - 100x - 75
At x= -5.00 F(x) is equal to -150.00
At x= -4.00 F(x) is equal to 21.00
Intuitively we feel, and justly so, that since F(x) is negative on one side of the interval, and positive on the other side then, somewhere inside this interval, F(x) 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 -150.000000000 -4.000000000 21.000000000
0.000000000 -75.000000000 -4.000000000 21.000000000
0.000000000 -75.000000000 -2.000000000 81.000000000
0.000000000 -75.000000000 -1.000000000 18.000000000
-0.500000000 -26.250000000 -1.000000000 18.000000000
-0.750000000 -3.375000000 -1.000000000 18.000000000
-0.750000000 -3.375000000 -0.875000000 7.523437500
-0.750000000 -3.375000000 -0.812500000 2.124023438
-0.781250000 -0.613403320 -0.812500000 2.124023438
-0.781250000 -0.613403320 -0.796875000 0.758377075
-0.781250000 -0.613403320 -0.789062500 0.073247910
-0.785156250 -0.269888163 -0.789062500 0.073247910
-0.787109375 -0.098272651 -0.789062500 0.073247910
-0.788085938 -0.012500491 -0.789062500 0.073247910
-0.788085938 -0.012500491 -0.788574219 0.030376681
-0.788085938 -0.012500491 -0.788330078 0.008938838
-0.788208008 -0.001780641 -0.788330078 0.008938838
-0.788208008 -0.001780641 -0.788269043 0.003579145
-0.788208008 -0.001780641 -0.788238525 0.000899263
-0.788223267 -0.000440686 -0.788238525 0.000899263
-0.788223267 -0.000440686 -0.788230896 0.000229289
-0.788227081 -0.000105698 -0.788230896 0.000229289
-0.788227081 -0.000105698 -0.788228989 0.000061796
-0.788228035 -0.000021951 -0.788228989 0.000061796
-0.788228035 -0.000021951 -0.788228512 0.000019922
-0.788228273 -0.000001014 -0.788228512 0.000019922
-0.788228273 -0.000001014 -0.788228393 0.000009454
-0.788228273 -0.000001014 -0.788228333 0.000004220
Next Middle will get us close enough to zero:
F( -0.788228288 ) is 0.000000294
The desired approximation of the solution is:
x ≓ -0.788228288
Note, ≓ is the approximation symbol
One solution was found :
x ≓ -0.788228288
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