Mister Exam
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Identical expressions
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Similar expressions
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x^3+9(x)^2+24x+17=0 equation
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The solution
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[src]
3 2 x + 9*x + 24*x + 17 = 0
$$\left(24 x + \left(x^{3} + 9 x^{2}\right)\right) + 17 = 0$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 9$$
$$q = \frac{c}{a}$$
$$q = 24$$
$$v = \frac{d}{a}$$
$$v = 17$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -9$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 24$$
$$x_{1} x_{2} x_{3} = 17$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 9$$
$$q = \frac{c}{a}$$
$$q = 24$$
$$v = \frac{d}{a}$$
$$v = 17$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -9$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 24$$
$$x_{1} x_{2} x_{3} = 17$$
Rapid solution
[src]
/pi\ / /pi\ ___ /pi\ \ ___ /pi\
cos|--| | sin|--| \/ 3 *cos|--| | \/ 3 *sin|--|
\9 / | \9 / \9 / / 1 \| \9 / / 1 \
x1 = -3 - ------- + I*|- ------- - ------------- + im|----------------------------------|| + ------------- + re|----------------------------------|
2 | 2 2 | _____________ || 2 | _____________ |
| | / ___ / ___\|| | / ___ / ___\|
| | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | 1 I*\/ 3 ||
| |3 / - + ------- *|- - - -------||| |3 / - + ------- *|- - - -------||
\ \\/ 2 2 \ 2 2 /// \\/ 2 2 \ 2 2 //
$$x_{1} = -3 - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right)$$
/pi\ / /pi\ ___ /pi\ \ ___ /pi\
cos|--| | sin|--| \/ 3 *cos|--| | \/ 3 *sin|--|
\9 / | \9 / \9 / / 1 \| \9 / / 1 \
x2 = -3 - ------- + I*|- ------- + ------------- + im|----------------------------------|| - ------------- + re|----------------------------------|
2 | 2 2 | _____________ || 2 | _____________ |
| | / ___ / ___\|| | / ___ / ___\|
| | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | 1 I*\/ 3 ||
| |3 / - + ------- *|- - + -------||| |3 / - + ------- *|- - + -------||
\ \\/ 2 2 \ 2 2 /// \\/ 2 2 \ 2 2 //
$$x_{2} = -3 + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)$$
_____________
/ ___
1 / 1 I*\/ 3
x3 = -3 + ------------------ + 3 / - + -------
_____________ \/ 2 2
/ ___
/ 1 I*\/ 3
3 / - + -------
\/ 2 2
$$x_{3} = -3 + \frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}$$
x3 = -3 + (1/2 + sqrt(3)*i/2)^(-1/3) + (1/2 + sqrt(3)*i/2)^(1/3)
Sum and product of roots
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sum
/pi\ / /pi\ ___ /pi\ \ ___ /pi\ /pi\ / /pi\ ___ /pi\ \ ___ /pi\ _____________
cos|--| | sin|--| \/ 3 *cos|--| | \/ 3 *sin|--| cos|--| | sin|--| \/ 3 *cos|--| | \/ 3 *sin|--| / ___
\9 / | \9 / \9 / / 1 \| \9 / / 1 \ \9 / | \9 / \9 / / 1 \| \9 / / 1 \ 1 / 1 I*\/ 3
-3 - ------- + I*|- ------- - ------------- + im|----------------------------------|| + ------------- + re|----------------------------------| + -3 - ------- + I*|- ------- + ------------- + im|----------------------------------|| - ------------- + re|----------------------------------| + -3 + ------------------ + 3 / - + -------
2 | 2 2 | _____________ || 2 | _____________ | 2 | 2 2 | _____________ || 2 | _____________ | _____________ \/ 2 2
| | / ___ / ___\|| | / ___ / ___\| | | / ___ / ___\|| | / ___ / ___\| / ___
| | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | 1 I*\/ 3 || | | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | 1 I*\/ 3 || / 1 I*\/ 3
| |3 / - + ------- *|- - - -------||| |3 / - + ------- *|- - - -------|| | |3 / - + ------- *|- - + -------||| |3 / - + ------- *|- - + -------|| 3 / - + -------
\ \\/ 2 2 \ 2 2 /// \\/ 2 2 \ 2 2 // \ \\/ 2 2 \ 2 2 /// \\/ 2 2 \ 2 2 // \/ 2 2
$$\left(\left(-3 + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\right) + \left(-3 - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right)\right)\right) + \left(-3 + \frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}\right)$$
=
_____________ / /pi\ ___ /pi\ \ / /pi\ ___ /pi\ \
/ ___ | sin|--| \/ 3 *cos|--| | | sin|--| \/ 3 *cos|--| |
1 / 1 I*\/ 3 /pi\ | \9 / \9 / / 1 \| | \9 / \9 / / 1 \| / 1 \ / 1 \
-9 + ------------------ + 3 / - + ------- - cos|--| + I*|- ------- + ------------- + im|----------------------------------|| + I*|- ------- - ------------- + im|----------------------------------|| + re|----------------------------------| + re|----------------------------------|
_____________ \/ 2 2 \9 / | 2 2 | _____________ || | 2 2 | _____________ || | _____________ | | _____________ |
/ ___ | | / ___ / ___\|| | | / ___ / ___\|| | / ___ / ___\| | / ___ / ___\|
/ 1 I*\/ 3 | | / 1 I*\/ 3 | 1 I*\/ 3 ||| | | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | 1 I*\/ 3 || | / 1 I*\/ 3 | 1 I*\/ 3 ||
3 / - + ------- | |3 / - + ------- *|- - + -------||| | |3 / - + ------- *|- - - -------||| |3 / - + ------- *|- - + -------|| |3 / - + ------- *|- - - -------||
\/ 2 2 \ \\/ 2 2 \ 2 2 /// \ \\/ 2 2 \ 2 2 /// \\/ 2 2 \ 2 2 // \\/ 2 2 \ 2 2 //
$$-9 - \cos{\left(\frac{\pi}{9} \right)} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right) + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right) + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}$$
product
/ /pi\ / /pi\ ___ /pi\ \ ___ /pi\ \ / /pi\ / /pi\ ___ /pi\ \ ___ /pi\ \ / _____________\ | cos|--| | sin|--| \/ 3 *cos|--| | \/ 3 *sin|--| | | cos|--| | sin|--| \/ 3 *cos|--| | \/ 3 *sin|--| | | / ___ | | \9 / | \9 / \9 / / 1 \| \9 / / 1 \| | \9 / | \9 / \9 / / 1 \| \9 / / 1 \| | 1 / 1 I*\/ 3 | |-3 - ------- + I*|- ------- - ------------- + im|----------------------------------|| + ------------- + re|----------------------------------||*|-3 - ------- + I*|- ------- + ------------- + im|----------------------------------|| - ------------- + re|----------------------------------||*|-3 + ------------------ + 3 / - + ------- | | 2 | 2 2 | _____________ || 2 | _____________ || | 2 | 2 2 | _____________ || 2 | _____________ || | _____________ \/ 2 2 | | | | / ___ / ___\|| | / ___ / ___\|| | | | / ___ / ___\|| | / ___ / ___\|| | / ___ | | | | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | 1 I*\/ 3 ||| | | | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | | | |3 / - + ------- *|- - - -------||| |3 / - + ------- *|- - - -------||| | | |3 / - + ------- *|- - + -------||| |3 / - + ------- *|- - + -------||| | 3 / - + ------- | \ \ \\/ 2 2 \ 2 2 /// \\/ 2 2 \ 2 2 /// \ \ \\/ 2 2 \ 2 2 /// \\/ 2 2 \ 2 2 /// \ \/ 2 2 /
$$\left(-3 - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right)\right) \left(-3 + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\right) \left(-3 + \frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}\right)$$
=
/ _____________ / _____________\\
/ /pi\\ | 3 ___ 3 / ___ | 2/3 3 / ___ || / ___ /pi\ /pi\\
|6 + 4*sin|--||*\2*\/ 2 + \/ 1 + I*\/ 3 *\-6 + 2 *\/ 1 + I*\/ 3 //*|3 + \/ 3 *sin|--| + cos|--||
\ \18// \ \9 / \9 //
-------------------------------------------------------------------------------------------------------
_____________
3 / ___
4*\/ 1 + I*\/ 3
$$\frac{\left(2 \sqrt[3]{2} + \left(-6 + 2^{\frac{2}{3}} \sqrt[3]{1 + \sqrt{3} i}\right) \sqrt[3]{1 + \sqrt{3} i}\right) \left(4 \sin{\left(\frac{\pi}{18} \right)} + 6\right) \left(\sqrt{3} \sin{\left(\frac{\pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} + 3\right)}{4 \sqrt[3]{1 + \sqrt{3} i}}$$
(6 + 4*sin(pi/18))*(2*2^(1/3) + (1 + i*sqrt(3))^(1/3)*(-6 + 2^(2/3)*(1 + i*sqrt(3))^(1/3)))*(3 + sqrt(3)*sin(pi/9) + cos(pi/9))/(4*(1 + i*sqrt(3))^(1/3))
Numerical answer
[src]
x1 = -1.12061475842818
x2 = -3.34729635533386
x3 = -4.53208888623796
x3 = -4.53208888623796