If you successfully guess one root of the cubic equation, you can factorize the cubic polynomial using the Factor Theorem and then solve the resulting quadratic equation easily.
33−22−9−4=0
3
x
3
−
2
x
2
−
9
x
−
4
=
0
On inspection, =−1
x
=
−
1
is found to satisfy the equation. Now the cubic polynomial can be factorized.
33−22−9−4=0
3
x
3
−
2
x
2
−
9
x
−
4
=
0
⟹(+1)(32−5−4)=0
⟹
(
x
+
1
)
(
3
x
2
−
5
x
−
4
)
=
0
One root is −1
−
1
. The rest at the roots of the following quadratic equation.
32−5−4=0
3
x
2
−
5
x
−
4
=
0
⟹=−(−5)±(−5)2−4×3×(−4)‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√2×3
⟹
x
=
−
(
−
5
)
±
(
−
5
)
2
−
4
×
3
×
(
−
4
)
2
×
3
⟹=5±25+48‾‾‾‾‾‾‾‾√6
⟹
x
=
5
±
25
+
48
6
⟹=5±73‾‾‾√6
⟹
x
=
5
±
73
6
⟹∈{−1,5−73‾‾‾√6,5+73‾‾‾√6}
⟹
x
∈
{
−
1
,
5
−
73
6
,
5
+
73
6
}
Some cubic equations to be solved by students are deliberately made to have one simple root which can be guessed.