Answer:
Explanation:
To solve the equation x^3 - 15x = 4 using Cardano's formula, we need to follow these steps:
1. Rewrite the equation in the form x^3 + px = q:
x^3 - 15x - 4 = 0
2. Identify the values of p and q from the equation:
p = -15
q = -4
3. Calculate the value of delta (Δ):
Δ = (q^2/4) + (p^3/27)
= (-4^2/4) + (-15^3/27)
= 1 + (-375/27)
= 1 - 13.89
= -12.89
4. Calculate the value of u:
u = cbrt((-q/2) + sqrt(Δ))
= cbrt((-(-4)/2) + sqrt(-12.89))
= cbrt(2 + sqrt(-12.89))
5. Calculate the value of v:
v = cbrt((-q/2) - sqrt(Δ))
= cbrt((-(-4)/2) - sqrt(-12.89))
= cbrt(2 - sqrt(-12.89))
6. Calculate the three solutions using Cardano's formula:
x1 = u + v
x2 = -(u + v)/2 + (u - v)*i*sqrt(3)/2
x3 = -(u + v)/2 - (u - v)*i*sqrt(3)/2
Note: Here, i represents the imaginary unit.
By substituting the calculated values of u and v, we can find the three solutions for the equation.