Mister Exam
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How to use it?
- Equation:
- Equation x-y=7
-
Equation 1/(x+6)=2
-
Equation (x-4)^2-x^2=0
-
Equation 6/(x+5)=-5
- Express {x} in terms of y where:
- 7*x-1*y=-7
- 10*x-15*y=16
- -14*x-12*y=5
- 13*x-12*y=19
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Identical expressions
- two x^ three + four x^2-3x+ six /4= zero
- 2x cubed plus 4x squared minus 3x plus 6 divide by 4 equally 0
- two x to the power of three plus four x squared minus 3x plus six divide by 4 equally zero
- 2x3+4x2-3x+6/4=0
- 2x³+4x²-3x+6/4=0
- 2x to the power of 3+4x to the power of 2-3x+6/4=0
- 2x^3+4x^2-3x+6/4=O
- 2x^3+4x^2-3x+6 divide by 4=0
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Similar expressions
- 2x^3+4x^2-3x-6/4=0
- 2x^3-4x^2-3x+6/4=0
- 2x^3+4x^2+3x+6/4=0
2x^3+4x^2-3x+6/4=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
3 2 3
2*x + 4*x - 3*x + - = 0
2
$$2 x^{3} + 4 x^{2} - 3 x + \frac{3}{2} = 0$$
Vieta's Theorem
rewrite the equation
$$2 x^{3} + 4 x^{2} - 3 x + \frac{3}{2} = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + 2 x^{2} - \frac{3 x}{2} + \frac{3}{4} = 0$$
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = 2$$
$$q = \frac{c}{a}$$
$$q = - \frac{3}{2}$$
$$v = \frac{d}{a}$$
$$v = \frac{3}{4}$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -2$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{3}{2}$$
$$x_{1} x_{2} x_{3} = \frac{3}{4}$$
$$2 x^{3} + 4 x^{2} - 3 x + \frac{3}{2} = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + 2 x^{2} - \frac{3 x}{2} + \frac{3}{4} = 0$$
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = 2$$
$$q = \frac{c}{a}$$
$$q = - \frac{3}{2}$$
$$v = \frac{d}{a}$$
$$v = \frac{3}{4}$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -2$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{3}{2}$$
$$x_{1} x_{2} x_{3} = \frac{3}{4}$$
Rapid solution
[src]
_________________ / 2/3\
3 / _____ ___ | / _____\ |
2 \/ 253 + 9*\/ 305 17 I*\/ 3 *\-34 + \253 + 9*\/ 305 / /
x_1 = - - + -------------------- + ---------------------- + ------------------------------------
3 12 _________________ _________________
3 / _____ 3 / _____
6*\/ 253 + 9*\/ 305 12*\/ 253 + 9*\/ 305
$$x_{1} = - \frac{2}{3} + \frac{17}{6 \sqrt[3]{9 \sqrt{305} + 253}} + \frac{\sqrt[3]{9 \sqrt{305} + 253}}{12} + \frac{\sqrt{3} i \left(-34 + \left(9 \sqrt{305} + 253\right)^{\frac{2}{3}}\right)}{12 \sqrt[3]{9 \sqrt{305} + 253}}$$
_________________ / 2/3\
3 / _____ ___ | / _____\ |
2 \/ 253 + 9*\/ 305 17 I*\/ 3 *\34 - \253 + 9*\/ 305 / /
x_2 = - - + -------------------- + ---------------------- + -----------------------------------
3 12 _________________ _________________
3 / _____ 3 / _____
6*\/ 253 + 9*\/ 305 12*\/ 253 + 9*\/ 305
$$x_{2} = - \frac{2}{3} + \frac{17}{6 \sqrt[3]{9 \sqrt{305} + 253}} + \frac{\sqrt[3]{9 \sqrt{305} + 253}}{12} + \frac{\sqrt{3} i \left(- \left(9 \sqrt{305} + 253\right)^{\frac{2}{3}} + 34\right)}{12 \sqrt[3]{9 \sqrt{305} + 253}}$$
_________________
3 / _____
2 17 \/ 253 + 9*\/ 305
x_3 = - - - ---------------------- - --------------------
3 _________________ 6
3 / _____
3*\/ 253 + 9*\/ 305
$$x_{3} = - \frac{\sqrt[3]{9 \sqrt{305} + 253}}{6} - \frac{17}{3 \sqrt[3]{9 \sqrt{305} + 253}} - \frac{2}{3}$$
Sum and product of roots
[src]
sum
_________________ / 2/3\ _________________ / 2/3\ _________________
3 / _____ ___ | / _____\ | 3 / _____ ___ | / _____\ | 3 / _____
2 \/ 253 + 9*\/ 305 17 I*\/ 3 *\-34 + \253 + 9*\/ 305 / / 2 \/ 253 + 9*\/ 305 17 I*\/ 3 *\34 - \253 + 9*\/ 305 / / 2 17 \/ 253 + 9*\/ 305
- - + -------------------- + ---------------------- + ------------------------------------ + - - + -------------------- + ---------------------- + ----------------------------------- + - - - ---------------------- - --------------------
3 12 _________________ _________________ 3 12 _________________ _________________ 3 _________________ 6
3 / _____ 3 / _____ 3 / _____ 3 / _____ 3 / _____
6*\/ 253 + 9*\/ 305 12*\/ 253 + 9*\/ 305 6*\/ 253 + 9*\/ 305 12*\/ 253 + 9*\/ 305 3*\/ 253 + 9*\/ 305
$$\left(- \frac{2}{3} + \frac{17}{6 \sqrt[3]{9 \sqrt{305} + 253}} + \frac{\sqrt[3]{9 \sqrt{305} + 253}}{12} + \frac{\sqrt{3} i \left(-34 + \left(9 \sqrt{305} + 253\right)^{\frac{2}{3}}\right)}{12 \sqrt[3]{9 \sqrt{305} + 253}}\right) + \left(- \frac{2}{3} + \frac{17}{6 \sqrt[3]{9 \sqrt{305} + 253}} + \frac{\sqrt[3]{9 \sqrt{305} + 253}}{12} + \frac{\sqrt{3} i \left(- \left(9 \sqrt{305} + 253\right)^{\frac{2}{3}} + 34\right)}{12 \sqrt[3]{9 \sqrt{305} + 253}}\right) + \left(- \frac{\sqrt[3]{9 \sqrt{305} + 253}}{6} - \frac{17}{3 \sqrt[3]{9 \sqrt{305} + 253}} - \frac{2}{3}\right)$$
=
/ 2/3\ / 2/3\
___ | / _____\ | ___ | / _____\ |
I*\/ 3 *\-34 + \253 + 9*\/ 305 / / I*\/ 3 *\34 - \253 + 9*\/ 305 / /
-2 + ------------------------------------ + -----------------------------------
_________________ _________________
3 / _____ 3 / _____
12*\/ 253 + 9*\/ 305 12*\/ 253 + 9*\/ 305
$$-2 + \frac{\sqrt{3} i \left(- \left(9 \sqrt{305} + 253\right)^{\frac{2}{3}} + 34\right)}{12 \sqrt[3]{9 \sqrt{305} + 253}} + \frac{\sqrt{3} i \left(-34 + \left(9 \sqrt{305} + 253\right)^{\frac{2}{3}}\right)}{12 \sqrt[3]{9 \sqrt{305} + 253}}$$
product
_________________ / 2/3\ _________________ / 2/3\ _________________
3 / _____ ___ | / _____\ | 3 / _____ ___ | / _____\ | 3 / _____
2 \/ 253 + 9*\/ 305 17 I*\/ 3 *\-34 + \253 + 9*\/ 305 / / 2 \/ 253 + 9*\/ 305 17 I*\/ 3 *\34 - \253 + 9*\/ 305 / / 2 17 \/ 253 + 9*\/ 305
- - + -------------------- + ---------------------- + ------------------------------------ * - - + -------------------- + ---------------------- + ----------------------------------- * - - - ---------------------- - --------------------
3 12 _________________ _________________ 3 12 _________________ _________________ 3 _________________ 6
3 / _____ 3 / _____ 3 / _____ 3 / _____ 3 / _____
6*\/ 253 + 9*\/ 305 12*\/ 253 + 9*\/ 305 6*\/ 253 + 9*\/ 305 12*\/ 253 + 9*\/ 305 3*\/ 253 + 9*\/ 305
$$\left(- \frac{2}{3} + \frac{17}{6 \sqrt[3]{9 \sqrt{305} + 253}} + \frac{\sqrt[3]{9 \sqrt{305} + 253}}{12} + \frac{\sqrt{3} i \left(-34 + \left(9 \sqrt{305} + 253\right)^{\frac{2}{3}}\right)}{12 \sqrt[3]{9 \sqrt{305} + 253}}\right) * \left(- \frac{2}{3} + \frac{17}{6 \sqrt[3]{9 \sqrt{305} + 253}} + \frac{\sqrt[3]{9 \sqrt{305} + 253}}{12} + \frac{\sqrt{3} i \left(- \left(9 \sqrt{305} + 253\right)^{\frac{2}{3}} + 34\right)}{12 \sqrt[3]{9 \sqrt{305} + 253}}\right) * \left(- \frac{\sqrt[3]{9 \sqrt{305} + 253}}{6} - \frac{17}{3 \sqrt[3]{9 \sqrt{305} + 253}} - \frac{2}{3}\right)$$
=
-3/4
$$- \frac{3}{4}$$
Numerical answer
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x1 = 0.33383805030548 - 0.411941419035783*i
x2 = 0.33383805030548 + 0.411941419035783*i
x3 = -2.66767610061096
x3 = -2.66767610061096
The graph