How to solve the cubic equation: #9x^3 + 3x^2 -23x +4 = 0# ?

1 Answer
Feb 27, 2018

# x = -1.84712709 " or " 0.18046042" or "4/3.#

Explanation:

#"Apply the rational roots theorem."#
#"We search for roots of the shape "pm p/q", with"#
#p" a divisor of 4 and "q" a divisor of 9."#
#"We find "x=4/3" as rational root."#
#"So "(3x - 4)" is a factor, we divide it away : "#
#9 x^3 + 3 x^2 - 23 x + 4 = (3 x - 4)(3 x^2 + 5 x - 1)#
#"Solving the remaining quadratic equation, gives the other roots :"#
#3 x^2 + 5 x - 1 = 0#
#"disc "5^2 + 4*3 = 37#
#=> x = (-5 pm sqrt(37))/6#
#=> x = -1.84712709 " or " 0.18046042.#