SOLUTION: Can you please solve this equation using Cardano's method: x^3+24x-25=0

Question 1095189: Can you please solve this equation using Cardano's method:
x^3+24x-25=0
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
The general form for a cubic equation is
:
ax^3 + bx^2 + cx + d = 0
:
the problem's cubic equation is
:
x^3 + 24x - 25 = 0
:
this is a depressed cubic equation, the general form is x^3 + Ax = B
:
therefore we have
:
x^3 + 24x = 25
:
we determine s and t so that
:
1) 3st = A
2) s^3 - t^3 = B
:
Cardano's method tells us that x = s - t will be one of the roots of the cubic equation
:
Solving the equation 1 for s and substituting into equation 2, we get
:
(A/3t)^3 - t^3 = B
:
simplifying this we get
:
t^6 + Bt^3 - (A^3/27) = 0
:
let u = t^3, then we have a quadratic equation
:
u^2 + Bu - (A^3/27) = 0
:
referring to our depressed cubic equation, we need s and t to satisfy
:
3) 3st = 24
4) s^3 - t^3 = 25
:
u^2 +25u -(24^3/27) = 0
:
using quadratic formula, we have
:
u = (-25 + square root(25^2 - 4 * 1 * (-512))) / (2*1) = 13.3505
u = (-25 - square root(25^2 - 4 * 1 * (-512))) / (2*1) = -38.3505
:
we reject the negative solution for u, therefore
:
t^3 = 13.3505
s^3 = t^3 + 25 = 13.3505 + 25 = 38.3508
:
*****************************************************************************
x = (38.3505)^(1/3) - (13.3505)^(1/3) = 1.000000955 approximately 1
:
therefore one solution is x=1, the other two solutions are complex
:
(x-1)(x^2+x+25) = 0 = x^3 + 24x - 25
:
we see that the two solutions to x^2 + x + 25 = 0 (using quadratic formula) are
:
x = (1/2)(-1-3isquare root(11)
x = (1/2)(-1+3isquare root(11)
*****************************************************************************
:
*********************************************
:
RELATED QUESTIONS
Use Cardano�s formula to solve x^3 = 6x +... (answered by solver91311)

Hello, can you please solve this equation using Cardano's method? x^3 + 12x + 32 = 0 (answered by ikleyn)

Can you Please Help Me Solve This equation by using the substitution method? -x+y+0... (answered by jim_thompson5910)

Can you please help me. I'm want to solve this system of equation using the substitution... (answered by josgarithmetic,ikleyn)

Can you please help me. I'm want to solve this system of equation using the substitution... (answered by Alan3354)

Could you please help me solve this quadratic equation using completing the square... (answered by Alan3354)

Could you please help solve this quadratic equation using completing the square method? (answered by lwsshak3,josgarithmetic,josmiceli)

please solve this problem on quadratic equation using factoring method (2x-3)^2 =... (answered by HyperBrain)

how can i solve this equation using substitution method? x+y=12... (answered by Fombitz)