Mister Exam
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- Equation:
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Equation 3x^2=2x+x^3
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Equation 4x^3-4x=0
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Equation -x=3,5
- Express {x} in terms of y where:
- -2*x-5*y=16
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- 6*x+10*y=-19
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Identical expressions
- two *x^ three - three *x^ two + thirty-six *x- eleven = zero
- 2 multiply by x cubed minus 3 multiply by x squared plus 36 multiply by x minus 11 equally 0
- two multiply by x to the power of three minus three multiply by x to the power of two plus thirty minus six multiply by x minus eleven equally zero
- 2*x3-3*x2+36*x-11=0
- 2*x³-3*x²+36*x-11=0
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- 2x^3-3x^2+36x-11=0
- 2x3-3x2+36x-11=0
- 2*x^3-3*x^2+36*x-11=O
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Similar expressions
- 2*x^3-3*x^2-36*x-11=0
- 2*x^3-3*x^2+36*x+11=0
- 2*x^3+3*x^2+36*x-11=0
2*x^3-3*x^2+36*x-11=0 equation
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The solution
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[src]
3 2 2*x - 3*x + 36*x - 11 = 0
$$\left(36 x + \left(2 x^{3} - 3 x^{2}\right)\right) - 11 = 0$$
Vieta's Theorem
rewrite the equation
$$\left(36 x + \left(2 x^{3} - 3 x^{2}\right)\right) - 11 = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{3 x^{2}}{2} + 18 x - \frac{11}{2} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{3}{2}$$
$$q = \frac{c}{a}$$
$$q = 18$$
$$v = \frac{d}{a}$$
$$v = - \frac{11}{2}$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{3}{2}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 18$$
$$x_{1} x_{2} x_{3} = - \frac{11}{2}$$
$$\left(36 x + \left(2 x^{3} - 3 x^{2}\right)\right) - 11 = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{3 x^{2}}{2} + 18 x - \frac{11}{2} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{3}{2}$$
$$q = \frac{c}{a}$$
$$q = 18$$
$$v = \frac{d}{a}$$
$$v = - \frac{11}{2}$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{3}{2}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 18$$
$$x_{1} x_{2} x_{3} = - \frac{11}{2}$$
Rapid solution
[src]
__________________ / __________________ \
/ _____ | / _____ |
/ 351 27*\/ 771 | ___ / 351 27*\/ 771 |
3 / --- + ---------- |\/ 3 *3 / --- + ---------- ___ |
1 69 \/ 8 2 | \/ 8 2 69*\/ 3 |
x1 = - - ------------------------- + ----------------------- + I*|----------------------------- + -------------------------|
2 __________________ 6 | 6 __________________|
/ _____ | / _____ |
/ 351 27*\/ 771 | / 351 27*\/ 771 |
8*3 / --- + ---------- | 8*3 / --- + ---------- |
\/ 8 2 \ \/ 8 2 /
$$x_{1} = - \frac{69}{8 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}} + \frac{1}{2} + \frac{\sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{6} + i \left(\frac{69 \sqrt{3}}{8 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{6}\right)$$
__________________ / __________________\
/ _____ | / _____ |
/ 351 27*\/ 771 | ___ / 351 27*\/ 771 |
3 / --- + ---------- | ___ \/ 3 *3 / --- + ---------- |
1 69 \/ 8 2 | 69*\/ 3 \/ 8 2 |
x2 = - - ------------------------- + ----------------------- + I*|- ------------------------- - -----------------------------|
2 __________________ 6 | __________________ 6 |
/ _____ | / _____ |
/ 351 27*\/ 771 | / 351 27*\/ 771 |
8*3 / --- + ---------- | 8*3 / --- + ---------- |
\/ 8 2 \ \/ 8 2 /
$$x_{2} = - \frac{69}{8 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}} + \frac{1}{2} + \frac{\sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{6} - \frac{69 \sqrt{3}}{8 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}\right)$$
__________________
/ _____
/ 351 27*\/ 771
3 / --- + ----------
1 \/ 8 2 69
x3 = - - ----------------------- + -------------------------
2 3 __________________
/ _____
/ 351 27*\/ 771
4*3 / --- + ----------
\/ 8 2
$$x_{3} = - \frac{\sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{3} + \frac{1}{2} + \frac{69}{4 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}$$
x3 = -(351/8 + 27*sqrt(771)/2)^(1/3)/3 + 1/2 + 69/(4*(351/8 + 27*sqrt(771)/2)^(1/3))
Sum and product of roots
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sum
__________________ / __________________ \ __________________ / __________________\ __________________
/ _____ | / _____ | / _____ | / _____ | / _____
/ 351 27*\/ 771 | ___ / 351 27*\/ 771 | / 351 27*\/ 771 | ___ / 351 27*\/ 771 | / 351 27*\/ 771
3 / --- + ---------- |\/ 3 *3 / --- + ---------- ___ | 3 / --- + ---------- | ___ \/ 3 *3 / --- + ---------- | 3 / --- + ----------
1 69 \/ 8 2 | \/ 8 2 69*\/ 3 | 1 69 \/ 8 2 | 69*\/ 3 \/ 8 2 | 1 \/ 8 2 69
- - ------------------------- + ----------------------- + I*|----------------------------- + -------------------------| + - - ------------------------- + ----------------------- + I*|- ------------------------- - -----------------------------| + - - ----------------------- + -------------------------
2 __________________ 6 | 6 __________________| 2 __________________ 6 | __________________ 6 | 2 3 __________________
/ _____ | / _____ | / _____ | / _____ | / _____
/ 351 27*\/ 771 | / 351 27*\/ 771 | / 351 27*\/ 771 | / 351 27*\/ 771 | / 351 27*\/ 771
8*3 / --- + ---------- | 8*3 / --- + ---------- | 8*3 / --- + ---------- | 8*3 / --- + ---------- | 4*3 / --- + ----------
\/ 8 2 \ \/ 8 2 / \/ 8 2 \ \/ 8 2 / \/ 8 2
$$\left(- \frac{\sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{3} + \frac{1}{2} + \frac{69}{4 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}\right) + \left(\left(- \frac{69}{8 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}} + \frac{1}{2} + \frac{\sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{6} - \frac{69 \sqrt{3}}{8 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}\right)\right) + \left(- \frac{69}{8 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}} + \frac{1}{2} + \frac{\sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{6} + i \left(\frac{69 \sqrt{3}}{8 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{6}\right)\right)\right)$$
=
/ __________________\ / __________________ \
| / _____ | | / _____ |
| ___ / 351 27*\/ 771 | | ___ / 351 27*\/ 771 |
| ___ \/ 3 *3 / --- + ---------- | |\/ 3 *3 / --- + ---------- ___ |
3 | 69*\/ 3 \/ 8 2 | | \/ 8 2 69*\/ 3 |
- + I*|- ------------------------- - -----------------------------| + I*|----------------------------- + -------------------------|
2 | __________________ 6 | | 6 __________________|
| / _____ | | / _____ |
| / 351 27*\/ 771 | | / 351 27*\/ 771 |
| 8*3 / --- + ---------- | | 8*3 / --- + ---------- |
\ \/ 8 2 / \ \/ 8 2 /
$$\frac{3}{2} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{6} - \frac{69 \sqrt{3}}{8 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}\right) + i \left(\frac{69 \sqrt{3}}{8 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{6}\right)$$
product
/ __________________ / __________________ \\ / __________________ / __________________\\ / __________________ \ | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 351 27*\/ 771 | ___ / 351 27*\/ 771 || | / 351 27*\/ 771 | ___ / 351 27*\/ 771 || | / 351 27*\/ 771 | | 3 / --- + ---------- |\/ 3 *3 / --- + ---------- ___ || | 3 / --- + ---------- | ___ \/ 3 *3 / --- + ---------- || | 3 / --- + ---------- | |1 69 \/ 8 2 | \/ 8 2 69*\/ 3 || |1 69 \/ 8 2 | 69*\/ 3 \/ 8 2 || |1 \/ 8 2 69 | |- - ------------------------- + ----------------------- + I*|----------------------------- + -------------------------||*|- - ------------------------- + ----------------------- + I*|- ------------------------- - -----------------------------||*|- - ----------------------- + -------------------------| |2 __________________ 6 | 6 __________________|| |2 __________________ 6 | __________________ 6 || |2 3 __________________| | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 351 27*\/ 771 | / 351 27*\/ 771 || | / 351 27*\/ 771 | / 351 27*\/ 771 || | / 351 27*\/ 771 | | 8*3 / --- + ---------- | 8*3 / --- + ---------- || | 8*3 / --- + ---------- | 8*3 / --- + ---------- || | 4*3 / --- + ---------- | \ \/ 8 2 \ \/ 8 2 // \ \/ 8 2 \ \/ 8 2 // \ \/ 8 2 /
$$\left(- \frac{69}{8 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}} + \frac{1}{2} + \frac{\sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{6} + i \left(\frac{69 \sqrt{3}}{8 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{6}\right)\right) \left(- \frac{69}{8 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}} + \frac{1}{2} + \frac{\sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{6} - \frac{69 \sqrt{3}}{8 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}{3} + \frac{1}{2} + \frac{69}{4 \sqrt[3]{\frac{351}{8} + \frac{27 \sqrt{771}}{2}}}\right)$$
=
11/2
$$\frac{11}{2}$$
11/2
Numerical answer
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x1 = 0.59401023643433 + 4.15650258915633*i
x2 = 0.31197952713134
x3 = 0.59401023643433 - 4.15650258915633*i
x3 = 0.59401023643433 - 4.15650258915633*i