Find a positive root of the cubic equation x35x70 using Card

Find a positive root of the cubic equation

x³+5x-7=0

using Cardan-Tartaglia method.

We are given the equation x³ + 5x - 7 = 0

Using the Cardan Tartaglia method, the solution to the equation x³+ ax²+ bx + c = 0 can be found as under:

Let x = u + v . Then this equation changes to u³ + v³ + (u + v)(3uv + p) + q = 0

where p = b – (a² /3) = 5 ( as a = 0 and b = 5) and q = c + (2a³ – 9ab)/ 27 = - 7 ( as a = 0 , b = 5 and c = - 7)

The solutions are u³ = (-q/2) + v(q² /4) + (p³ /27) = 7/2 + v ( 49/4) + 125/27 and

v³ = (-q/2) - v(q² /4) + (p³ /27) = 7/2 – v ( 49/4) + 125/27

Since 27q² + 4p³ = 27 (49) + 4 (125) = 1323 + 500 = 1823 which is > 0, the roots will be real.

We can find the cube roots of these equations to solve for u and v, by using the following formulae:

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a - b)³ = a³ - 3a²b + 3 ab² - b³

Then the solution is x = u + v.