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Identical expressions
- a*x^ three +b*x^ two +c*x+d= zero
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-
Similar expressions
- a*x^3-b*x^2+c*x+d=0
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- a*x^3+b*x^2-c*x+d=0
a*x^3+b*x^2+c*x+d=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Vieta's Theorem
rewrite the equation
$$a x^{3} + b x^{2} + c x + d = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$\frac{a x^{3} + b x^{2} + c x + d}{a} = 0$$
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{b}{a}$$
$$q = \frac{c}{a}$$
$$q = \frac{c}{a}$$
$$v = \frac{d}{a}$$
$$v = \frac{d}{a}$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \frac{b}{a}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{c}{a}$$
$$x_{1} x_{2} x_{3} = \frac{d}{a}$$
$$a x^{3} + b x^{2} + c x + d = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$\frac{a x^{3} + b x^{2} + c x + d}{a} = 0$$
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{b}{a}$$
$$q = \frac{c}{a}$$
$$q = \frac{c}{a}$$
$$v = \frac{d}{a}$$
$$v = \frac{d}{a}$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \frac{b}{a}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{c}{a}$$
$$x_{1} x_{2} x_{3} = \frac{d}{a}$$
The graph
Rapid solution
[src]
_____________________________________________________________________
/ ________________________________________
/ / 2 3
/ / / 3 \ / 2 \
/ / |2*b 27*d 9*b*c| |b 3*c|
/ / |---- + ---- - -----| - 4*|-- - ---|
/ / | 3 a 2 | | 2 a | 3 2
/ \/ \ a a / \a / b 27*d 9*b*c b 3*c
/ ----------------------------------------------- + -- + ---- - ----- -- - ---
3 / 2 3 2*a 2 2 a
\/ a 2*a b a
x_1 = - -------------------------------------------------------------------------------- - --- - ----------------------------------------------------------------------------------
3 3*a _____________________________________________________________________
/ ________________________________________
/ / 2 3
/ / / 3 \ / 2 \
/ / |2*b 27*d 9*b*c| |b 3*c|
/ / |---- + ---- - -----| - 4*|-- - ---|
/ / | 3 a 2 | | 2 a | 3
/ \/ \ a a / \a / b 27*d 9*b*c
3* / ----------------------------------------------- + -- + ---- - -----
3 / 2 3 2*a 2
\/ a 2*a
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}}{3} - \frac{- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}}{3 \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}} - \frac{b}{3 a}$$
_____________________________________________________________________
/ ________________________________________
/ / 2 3
/ / / 3 \ / 2 \
/ / |2*b 27*d 9*b*c| |b 3*c|
/ / |---- + ---- - -----| - 4*|-- - ---|
/ / | 3 a 2 | | 2 a | 3 / ___\ 2
/ \/ \ a a / \a / b 27*d 9*b*c | 1 I*\/ 3 | b 3*c
/ ----------------------------------------------- + -- + ---- - ----- *|- - - -------| -- - ---
3 / 2 3 2*a 2 \ 2 2 / 2 a
b \/ a 2*a a
x_2 = - --- - ------------------------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------
3*a 3 _____________________________________________________________________
/ ________________________________________
/ / 2 3
/ / / 3 \ / 2 \
/ / |2*b 27*d 9*b*c| |b 3*c|
/ / |---- + ---- - -----| - 4*|-- - ---|
/ ___\ / / | 3 a 2 | | 2 a | 3
| 1 I*\/ 3 | / \/ \ a a / \a / b 27*d 9*b*c
3*|- - - -------|* / ----------------------------------------------- + -- + ---- - -----
\ 2 2 / 3 / 2 3 2*a 2
\/ a 2*a
$$x_{2} = - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}}{3} - \frac{- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}} - \frac{b}{3 a}$$
_____________________________________________________________________
/ ________________________________________
/ / 2 3
/ / / 3 \ / 2 \
/ / |2*b 27*d 9*b*c| |b 3*c|
/ / |---- + ---- - -----| - 4*|-- - ---|
/ / | 3 a 2 | | 2 a | 3 / ___\ 2
/ \/ \ a a / \a / b 27*d 9*b*c | 1 I*\/ 3 | b 3*c
/ ----------------------------------------------- + -- + ---- - ----- *|- - + -------| -- - ---
3 / 2 3 2*a 2 \ 2 2 / 2 a
b \/ a 2*a a
x_3 = - --- - ------------------------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------
3*a 3 _____________________________________________________________________
/ ________________________________________
/ / 2 3
/ / / 3 \ / 2 \
/ / |2*b 27*d 9*b*c| |b 3*c|
/ / |---- + ---- - -----| - 4*|-- - ---|
/ ___\ / / | 3 a 2 | | 2 a | 3
| 1 I*\/ 3 | / \/ \ a a / \a / b 27*d 9*b*c
3*|- - + -------|* / ----------------------------------------------- + -- + ---- - -----
\ 2 2 / 3 / 2 3 2*a 2
\/ a 2*a
$$x_{3} = - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}}{3} - \frac{- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}} - \frac{b}{3 a}$$
Sum and product of roots
[src]
sum
_____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________
/ ________________________________________ / ________________________________________ / ________________________________________
/ / 2 3 / / 2 3 / / 2 3
/ / / 3 \ / 2 \ / / / 3 \ / 2 \ / / / 3 \ / 2 \
/ / |2*b 27*d 9*b*c| |b 3*c| / / |2*b 27*d 9*b*c| |b 3*c| / / |2*b 27*d 9*b*c| |b 3*c|
/ / |---- + ---- - -----| - 4*|-- - ---| / / |---- + ---- - -----| - 4*|-- - ---| / / |---- + ---- - -----| - 4*|-- - ---|
/ / | 3 a 2 | | 2 a | 3 2 / / | 3 a 2 | | 2 a | 3 / ___\ 2 / / | 3 a 2 | | 2 a | 3 / ___\ 2
/ \/ \ a a / \a / b 27*d 9*b*c b 3*c / \/ \ a a / \a / b 27*d 9*b*c | 1 I*\/ 3 | b 3*c / \/ \ a a / \a / b 27*d 9*b*c | 1 I*\/ 3 | b 3*c
/ ----------------------------------------------- + -- + ---- - ----- -- - --- / ----------------------------------------------- + -- + ---- - ----- *|- - - -------| -- - --- / ----------------------------------------------- + -- + ---- - ----- *|- - + -------| -- - ---
3 / 2 3 2*a 2 2 a 3 / 2 3 2*a 2 \ 2 2 / 2 a 3 / 2 3 2*a 2 \ 2 2 / 2 a
\/ a 2*a b a b \/ a 2*a a b \/ a 2*a a
- -------------------------------------------------------------------------------- - --- - ---------------------------------------------------------------------------------- + - --- - ------------------------------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------- + - --- - ------------------------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------
3 3*a _____________________________________________________________________ 3*a 3 _____________________________________________________________________ 3*a 3 _____________________________________________________________________
/ ________________________________________ / ________________________________________ / ________________________________________
/ / 2 3 / / 2 3 / / 2 3
/ / / 3 \ / 2 \ / / / 3 \ / 2 \ / / / 3 \ / 2 \
/ / |2*b 27*d 9*b*c| |b 3*c| / / |2*b 27*d 9*b*c| |b 3*c| / / |2*b 27*d 9*b*c| |b 3*c|
/ / |---- + ---- - -----| - 4*|-- - ---| / / |---- + ---- - -----| - 4*|-- - ---| / / |---- + ---- - -----| - 4*|-- - ---|
/ / | 3 a 2 | | 2 a | 3 / ___\ / / | 3 a 2 | | 2 a | 3 / ___\ / / | 3 a 2 | | 2 a | 3
/ \/ \ a a / \a / b 27*d 9*b*c | 1 I*\/ 3 | / \/ \ a a / \a / b 27*d 9*b*c | 1 I*\/ 3 | / \/ \ a a / \a / b 27*d 9*b*c
3* / ----------------------------------------------- + -- + ---- - ----- 3*|- - - -------|* / ----------------------------------------------- + -- + ---- - ----- 3*|- - + -------|* / ----------------------------------------------- + -- + ---- - -----
3 / 2 3 2*a 2 \ 2 2 / 3 / 2 3 2*a 2 \ 2 2 / 3 / 2 3 2*a 2
\/ a 2*a \/ a 2*a \/ a 2*a
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}}{3} - \frac{- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}}{3 \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}} - \frac{b}{3 a}\right) + \left(- \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}}{3} - \frac{- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}} - \frac{b}{3 a}\right) + \left(- \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}}{3} - \frac{- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}} - \frac{b}{3 a}\right)$$
=
_____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________
/ ________________________________________ / ________________________________________ / ________________________________________
/ / 2 3 / / 2 3 / / 2 3
/ / / 3 \ / 2 \ / / / 3 \ / 2 \ / / / 3 \ / 2 \
/ / |2*b 27*d 9*b*c| |b 3*c| / / |2*b 27*d 9*b*c| |b 3*c| / / |2*b 27*d 9*b*c| |b 3*c|
/ / |---- + ---- - -----| - 4*|-- - ---| / / |---- + ---- - -----| - 4*|-- - ---| / / |---- + ---- - -----| - 4*|-- - ---|
/ / | 3 a 2 | | 2 a | 3 2 / / | 3 a 2 | | 2 a | 3 / ___\ / / | 3 a 2 | | 2 a | 3 / ___\ 2 2
/ \/ \ a a / \a / b 27*d 9*b*c b 3*c / \/ \ a a / \a / b 27*d 9*b*c | 1 I*\/ 3 | / \/ \ a a / \a / b 27*d 9*b*c | 1 I*\/ 3 | b 3*c b 3*c
/ ----------------------------------------------- + -- + ---- - ----- -- - --- / ----------------------------------------------- + -- + ---- - ----- *|- - + -------| / ----------------------------------------------- + -- + ---- - ----- *|- - - -------| -- - --- -- - ---
3 / 2 3 2*a 2 2 a 3 / 2 3 2*a 2 \ 2 2 / 3 / 2 3 2*a 2 \ 2 2 / 2 a 2 a
\/ a 2*a b a \/ a 2*a \/ a 2*a a a
- -------------------------------------------------------------------------------- - - - ---------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------ - ------------------------------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------
3 a _____________________________________________________________________ 3 3 _____________________________________________________________________ _____________________________________________________________________
/ ________________________________________ / ________________________________________ / ________________________________________
/ / 2 3 / / 2 3 / / 2 3
/ / / 3 \ / 2 \ / / / 3 \ / 2 \ / / / 3 \ / 2 \
/ / |2*b 27*d 9*b*c| |b 3*c| / / |2*b 27*d 9*b*c| |b 3*c| / / |2*b 27*d 9*b*c| |b 3*c|
/ / |---- + ---- - -----| - 4*|-- - ---| / / |---- + ---- - -----| - 4*|-- - ---| / / |---- + ---- - -----| - 4*|-- - ---|
/ / | 3 a 2 | | 2 a | 3 / ___\ / / | 3 a 2 | | 2 a | 3 / ___\ / / | 3 a 2 | | 2 a | 3
/ \/ \ a a / \a / b 27*d 9*b*c | 1 I*\/ 3 | / \/ \ a a / \a / b 27*d 9*b*c | 1 I*\/ 3 | / \/ \ a a / \a / b 27*d 9*b*c
3* / ----------------------------------------------- + -- + ---- - ----- 3*|- - + -------|* / ----------------------------------------------- + -- + ---- - ----- 3*|- - - -------|* / ----------------------------------------------- + -- + ---- - -----
3 / 2 3 2*a 2 \ 2 2 / 3 / 2 3 2*a 2 \ 2 2 / 3 / 2 3 2*a 2
\/ a 2*a \/ a 2*a \/ a 2*a
$$- \frac{\sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}}{3} - \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}}{3} - \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}}{3} - \frac{- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}}{3 \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}} - \frac{- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}} - \frac{- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}} - \frac{b}{a}$$
product
_____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________
/ ________________________________________ / ________________________________________ / ________________________________________
/ / 2 3 / / 2 3 / / 2 3
/ / / 3 \ / 2 \ / / / 3 \ / 2 \ / / / 3 \ / 2 \
/ / |2*b 27*d 9*b*c| |b 3*c| / / |2*b 27*d 9*b*c| |b 3*c| / / |2*b 27*d 9*b*c| |b 3*c|
/ / |---- + ---- - -----| - 4*|-- - ---| / / |---- + ---- - -----| - 4*|-- - ---| / / |---- + ---- - -----| - 4*|-- - ---|
/ / | 3 a 2 | | 2 a | 3 2 / / | 3 a 2 | | 2 a | 3 / ___\ 2 / / | 3 a 2 | | 2 a | 3 / ___\ 2
/ \/ \ a a / \a / b 27*d 9*b*c b 3*c / \/ \ a a / \a / b 27*d 9*b*c | 1 I*\/ 3 | b 3*c / \/ \ a a / \a / b 27*d 9*b*c | 1 I*\/ 3 | b 3*c
/ ----------------------------------------------- + -- + ---- - ----- -- - --- / ----------------------------------------------- + -- + ---- - ----- *|- - - -------| -- - --- / ----------------------------------------------- + -- + ---- - ----- *|- - + -------| -- - ---
3 / 2 3 2*a 2 2 a 3 / 2 3 2*a 2 \ 2 2 / 2 a 3 / 2 3 2*a 2 \ 2 2 / 2 a
\/ a 2*a b a b \/ a 2*a a b \/ a 2*a a
- -------------------------------------------------------------------------------- - --- - ---------------------------------------------------------------------------------- * - --- - ------------------------------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------- * - --- - ------------------------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------
3 3*a _____________________________________________________________________ 3*a 3 _____________________________________________________________________ 3*a 3 _____________________________________________________________________
/ ________________________________________ / ________________________________________ / ________________________________________
/ / 2 3 / / 2 3 / / 2 3
/ / / 3 \ / 2 \ / / / 3 \ / 2 \ / / / 3 \ / 2 \
/ / |2*b 27*d 9*b*c| |b 3*c| / / |2*b 27*d 9*b*c| |b 3*c| / / |2*b 27*d 9*b*c| |b 3*c|
/ / |---- + ---- - -----| - 4*|-- - ---| / / |---- + ---- - -----| - 4*|-- - ---| / / |---- + ---- - -----| - 4*|-- - ---|
/ / | 3 a 2 | | 2 a | 3 / ___\ / / | 3 a 2 | | 2 a | 3 / ___\ / / | 3 a 2 | | 2 a | 3
/ \/ \ a a / \a / b 27*d 9*b*c | 1 I*\/ 3 | / \/ \ a a / \a / b 27*d 9*b*c | 1 I*\/ 3 | / \/ \ a a / \a / b 27*d 9*b*c
3* / ----------------------------------------------- + -- + ---- - ----- 3*|- - - -------|* / ----------------------------------------------- + -- + ---- - ----- 3*|- - + -------|* / ----------------------------------------------- + -- + ---- - -----
3 / 2 3 2*a 2 \ 2 2 / 3 / 2 3 2*a 2 \ 2 2 / 3 / 2 3 2*a 2
\/ a 2*a \/ a 2*a \/ a 2*a
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}}{3} - \frac{- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}}{3 \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}} - \frac{b}{3 a}\right) * \left(- \frac{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}}{3} - \frac{- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}}{3 \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}} - \frac{b}{3 a}\right) * \left(- \frac{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}}{3} - \frac{- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}}{3 \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{\sqrt{- 4 \left(- \frac{3 c}{a} + \frac{b^{2}}{a^{2}}\right)^{3} + \left(\frac{27 d}{a} - \frac{9 b c}{a^{2}} + \frac{2 b^{3}}{a^{3}}\right)^{2}}}{2} + \frac{27 d}{2 a} - \frac{9 b c}{2 a^{2}} + \frac{b^{3}}{a^{3}}}} - \frac{b}{3 a}\right)$$
=
-d --- a
$$- \frac{d}{a}$$