Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^2+3x+2=0
-
Equation z^2-6*z+10=0
-
Equation x^2+5x+6=0
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Equation 1*(x-1)*(x-3)=0
- Express {x} in terms of y where:
- -2*x+20*y=-2
- -20*x-6*y=4
- 6*x+9*y=-9
- -20*x-13*y=15
-
Identical expressions
- - two *x^ three - three *x^ two + eighteen = zero
- minus 2 multiply by x cubed minus 3 multiply by x squared plus 18 equally 0
- minus two multiply by x to the power of three minus three multiply by x to the power of two plus eighteen equally zero
- -2*x3-3*x2+18=0
- -2*x³-3*x²+18=0
- -2*x to the power of 3-3*x to the power of 2+18=0
- -2x^3-3x^2+18=0
- -2x3-3x2+18=0
- -2*x^3-3*x^2+18=O
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Similar expressions
- -2*x^3+3*x^2+18=0
- 2*x^3-3*x^2+18=0
- -2*x^3-3*x^2-18=0
-2*x^3-3*x^2+18=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Vieta's Theorem
rewrite the equation
$$- 2 x^{3} - 3 x^{2} + 18 = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{3 x^{2}}{2} - 9 = 0$$
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{3}{2}$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -9$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \frac{3}{2}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = -9$$
$$- 2 x^{3} - 3 x^{2} + 18 = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{3 x^{2}}{2} - 9 = 0$$
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{3}{2}$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -9$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \frac{3}{2}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = -9$$
Rapid solution
[src]
_______________ / 2/3\
3 / ____ ___ | / ____\ |
1 1 \/ 35 + 6*\/ 34 I*\/ 3 *\1 - \35 + 6*\/ 34 / /
x_1 = - - - -------------------- - ------------------ + --------------------------------
2 _______________ 4 _______________
3 / ____ 3 / ____
4*\/ 35 + 6*\/ 34 4*\/ 35 + 6*\/ 34
$$x_{1} = - \frac{\sqrt[3]{6 \sqrt{34} + 35}}{4} - \frac{1}{2} - \frac{1}{4 \sqrt[3]{6 \sqrt{34} + 35}} + \frac{\sqrt{3} i \left(- \left(6 \sqrt{34} + 35\right)^{\frac{2}{3}} + 1\right)}{4 \sqrt[3]{6 \sqrt{34} + 35}}$$
_______________ / 2/3\
3 / ____ ___ | / ____\ |
1 1 \/ 35 + 6*\/ 34 I*\/ 3 *\-1 + \35 + 6*\/ 34 / /
x_2 = - - - -------------------- - ------------------ + ---------------------------------
2 _______________ 4 _______________
3 / ____ 3 / ____
4*\/ 35 + 6*\/ 34 4*\/ 35 + 6*\/ 34
$$x_{2} = - \frac{\sqrt[3]{6 \sqrt{34} + 35}}{4} - \frac{1}{2} - \frac{1}{4 \sqrt[3]{6 \sqrt{34} + 35}} + \frac{\sqrt{3} i \left(-1 + \left(6 \sqrt{34} + 35\right)^{\frac{2}{3}}\right)}{4 \sqrt[3]{6 \sqrt{34} + 35}}$$
_______________
3 / ____
1 1 \/ 35 + 6*\/ 34
x_3 = - - + -------------------- + ------------------
2 _______________ 2
3 / ____
2*\/ 35 + 6*\/ 34
$$x_{3} = - \frac{1}{2} + \frac{1}{2 \sqrt[3]{6 \sqrt{34} + 35}} + \frac{\sqrt[3]{6 \sqrt{34} + 35}}{2}$$
Sum and product of roots
[src]
sum
_______________ / 2/3\ _______________ / 2/3\ _______________
3 / ____ ___ | / ____\ | 3 / ____ ___ | / ____\ | 3 / ____
1 1 \/ 35 + 6*\/ 34 I*\/ 3 *\1 - \35 + 6*\/ 34 / / 1 1 \/ 35 + 6*\/ 34 I*\/ 3 *\-1 + \35 + 6*\/ 34 / / 1 1 \/ 35 + 6*\/ 34
- - - -------------------- - ------------------ + -------------------------------- + - - - -------------------- - ------------------ + --------------------------------- + - - + -------------------- + ------------------
2 _______________ 4 _______________ 2 _______________ 4 _______________ 2 _______________ 2
3 / ____ 3 / ____ 3 / ____ 3 / ____ 3 / ____
4*\/ 35 + 6*\/ 34 4*\/ 35 + 6*\/ 34 4*\/ 35 + 6*\/ 34 4*\/ 35 + 6*\/ 34 2*\/ 35 + 6*\/ 34
$$\left(- \frac{\sqrt[3]{6 \sqrt{34} + 35}}{4} - \frac{1}{2} - \frac{1}{4 \sqrt[3]{6 \sqrt{34} + 35}} + \frac{\sqrt{3} i \left(- \left(6 \sqrt{34} + 35\right)^{\frac{2}{3}} + 1\right)}{4 \sqrt[3]{6 \sqrt{34} + 35}}\right) + \left(- \frac{\sqrt[3]{6 \sqrt{34} + 35}}{4} - \frac{1}{2} - \frac{1}{4 \sqrt[3]{6 \sqrt{34} + 35}} + \frac{\sqrt{3} i \left(-1 + \left(6 \sqrt{34} + 35\right)^{\frac{2}{3}}\right)}{4 \sqrt[3]{6 \sqrt{34} + 35}}\right) + \left(- \frac{1}{2} + \frac{1}{2 \sqrt[3]{6 \sqrt{34} + 35}} + \frac{\sqrt[3]{6 \sqrt{34} + 35}}{2}\right)$$
=
/ 2/3\ / 2/3\
___ | / ____\ | ___ | / ____\ |
3 I*\/ 3 *\1 - \35 + 6*\/ 34 / / I*\/ 3 *\-1 + \35 + 6*\/ 34 / /
- - + -------------------------------- + ---------------------------------
2 _______________ _______________
3 / ____ 3 / ____
4*\/ 35 + 6*\/ 34 4*\/ 35 + 6*\/ 34
$$- \frac{3}{2} + \frac{\sqrt{3} i \left(- \left(6 \sqrt{34} + 35\right)^{\frac{2}{3}} + 1\right)}{4 \sqrt[3]{6 \sqrt{34} + 35}} + \frac{\sqrt{3} i \left(-1 + \left(6 \sqrt{34} + 35\right)^{\frac{2}{3}}\right)}{4 \sqrt[3]{6 \sqrt{34} + 35}}$$
product
_______________ / 2/3\ _______________ / 2/3\ _______________
3 / ____ ___ | / ____\ | 3 / ____ ___ | / ____\ | 3 / ____
1 1 \/ 35 + 6*\/ 34 I*\/ 3 *\1 - \35 + 6*\/ 34 / / 1 1 \/ 35 + 6*\/ 34 I*\/ 3 *\-1 + \35 + 6*\/ 34 / / 1 1 \/ 35 + 6*\/ 34
- - - -------------------- - ------------------ + -------------------------------- * - - - -------------------- - ------------------ + --------------------------------- * - - + -------------------- + ------------------
2 _______________ 4 _______________ 2 _______________ 4 _______________ 2 _______________ 2
3 / ____ 3 / ____ 3 / ____ 3 / ____ 3 / ____
4*\/ 35 + 6*\/ 34 4*\/ 35 + 6*\/ 34 4*\/ 35 + 6*\/ 34 4*\/ 35 + 6*\/ 34 2*\/ 35 + 6*\/ 34
$$\left(- \frac{\sqrt[3]{6 \sqrt{34} + 35}}{4} - \frac{1}{2} - \frac{1}{4 \sqrt[3]{6 \sqrt{34} + 35}} + \frac{\sqrt{3} i \left(- \left(6 \sqrt{34} + 35\right)^{\frac{2}{3}} + 1\right)}{4 \sqrt[3]{6 \sqrt{34} + 35}}\right) * \left(- \frac{\sqrt[3]{6 \sqrt{34} + 35}}{4} - \frac{1}{2} - \frac{1}{4 \sqrt[3]{6 \sqrt{34} + 35}} + \frac{\sqrt{3} i \left(-1 + \left(6 \sqrt{34} + 35\right)^{\frac{2}{3}}\right)}{4 \sqrt[3]{6 \sqrt{34} + 35}}\right) * \left(- \frac{1}{2} + \frac{1}{2 \sqrt[3]{6 \sqrt{34} + 35}} + \frac{\sqrt[3]{6 \sqrt{34} + 35}}{2}\right)$$
=
9
$$9$$
Numerical answer
[src]
x1 = -1.59091603154739 - 1.67937290786214*i
x2 = 1.68183206309478
x3 = -1.59091603154739 + 1.67937290786214*i
x3 = -1.59091603154739 + 1.67937290786214*i
The graph