Solve x^3 -3x^2 +5x +4 = 0 using Tartaglia's method.
To solve the equation x^3 - 3x^2 + 5x + 4 = 0 using Tartaglia's method, also known as Cardano's method, we follow these steps:
Step 1: Identify the coefficients of the equation.
In this case, the coefficients are:
a = 1
b = -3
c = 5
d = 4
Step 2: Calculate the quantity Q.
Q = (3ac - b^2)/(9a^2)
= (3 * 1 * 5 - (-3)^2)/(9 * 1^2)
= (15 - 9)/(9)
= 6/9
= 2/3
Step 3: Calculate the quantity R.
R = (9abc - 27a^2d - 2b^3)/(54a^3)
= (9 * 1 * (-3) * 5 - 27 * 1^2 * 4 - 2 * (-3)^3)/(54 * 1^3)
= (-135 - 108 + 54)/(54)
= -189/54
= -7/2
Step 4: Calculate the quantity S.
S = (Q^3 + R^2)^(1/2)
= ((2/3)^3 + (-7/2)^2)^(1/2)
= ((8/27) + (49/4))^(1/2)
= (8/27 + 49/4)^(1/2)
= (128/108 + 588/108)^(1/2)
= (716/108)^(1/2)
≈ 2.562
Step 5: Find the values of u and v.
u = (R + S)^(1/3)
= (-7/2 + 2.562)^(1/3)
≈ (-2.438)^(1/3)
≈ -1.316
v = (R - S)^(1/3)
= (-7/2 - 2.562)^(1/3)
≈ (-4.062)^(1/3)
≈ -1.691
Step 6: Solve for x.
x = u - (b/(3a))
= -1.316 - (-3/(3 * 1))
= -1.316 + 1
≈ -0.316
x = v - (b/(3a))
= -1.691 - (-3/(3 * 1))
= -1.691 + 1
≈ -0.691
Therefore, the solutions to the equation x^3 - 3x^2 + 5x + 4 = 0 using Tartaglia's method are approximately -0.316 and -0.691.