Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 2x=18-x
-
Equation x^4=(2*x-3)^2
-
Equation 9*x^2=54*x
-
Equation cos(x)+sin(x)=0
- Express {x} in terms of y where:
- -18*x-1*y=-11
- 7*x-2*y=20
- -1*x-20*y=-17
- -2*x-14*y=7
- Factor polynomial:
- x^3+y^3
- Plot:
-
x^3+y^3
- Limit of the function:
- x^3+y^3
-
Identical expressions
- x^ three +y^ three
- x cubed plus y cubed
- x to the power of three plus y to the power of three
- x3+y3
- x³+y³
- x to the power of 3+y to the power of 3
-
Similar expressions
- x^3-y^3
x^3+y^3 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Vieta's Theorem
it is reduced cubic equation
$$p y^{2} + q y + v + y^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = x^{3}$$
Vieta Formulas
$$y_{1} + y_{2} + y_{3} = - p$$
$$y_{1} y_{2} + y_{1} y_{3} + y_{2} y_{3} = q$$
$$y_{1} y_{2} y_{3} = v$$
$$y_{1} + y_{2} + y_{3} = 0$$
$$y_{1} y_{2} + y_{1} y_{3} + y_{2} y_{3} = 0$$
$$y_{1} y_{2} y_{3} = x^{3}$$
$$p y^{2} + q y + v + y^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = x^{3}$$
Vieta Formulas
$$y_{1} + y_{2} + y_{3} = - p$$
$$y_{1} y_{2} + y_{1} y_{3} + y_{2} y_{3} = q$$
$$y_{1} y_{2} y_{3} = v$$
$$y_{1} + y_{2} + y_{3} = 0$$
$$y_{1} y_{2} + y_{1} y_{3} + y_{2} y_{3} = 0$$
$$y_{1} y_{2} y_{3} = x^{3}$$
The graph
Rapid solution
[src]
y1 = -re(x) - I*im(x)
$$y_{1} = - \operatorname{re}{\left(x\right)} - i \operatorname{im}{\left(x\right)}$$
/ ___ \ ___
re(x) |im(x) \/ 3 *re(x)| \/ 3 *im(x)
y2 = ----- + I*|----- - -----------| + -----------
2 \ 2 2 / 2
$$y_{2} = i \left(- \frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \frac{\operatorname{re}{\left(x\right)}}{2} + \frac{\sqrt{3} \operatorname{im}{\left(x\right)}}{2}$$
/ ___ \ ___
re(x) |im(x) \/ 3 *re(x)| \/ 3 *im(x)
y3 = ----- + I*|----- + -----------| - -----------
2 \ 2 2 / 2
$$y_{3} = i \left(\frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \frac{\operatorname{re}{\left(x\right)}}{2} - \frac{\sqrt{3} \operatorname{im}{\left(x\right)}}{2}$$
y3 = i*(sqrt(3)*re(x)/2 + im(x)/2) + re(x)/2 - sqrt(3)*im(x)/2
Sum and product of roots
[src]
sum
/ ___ \ ___ / ___ \ ___
re(x) |im(x) \/ 3 *re(x)| \/ 3 *im(x) re(x) |im(x) \/ 3 *re(x)| \/ 3 *im(x)
-re(x) - I*im(x) + ----- + I*|----- - -----------| + ----------- + ----- + I*|----- + -----------| - -----------
2 \ 2 2 / 2 2 \ 2 2 / 2
$$\left(\left(- \operatorname{re}{\left(x\right)} - i \operatorname{im}{\left(x\right)}\right) + \left(i \left(- \frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \frac{\operatorname{re}{\left(x\right)}}{2} + \frac{\sqrt{3} \operatorname{im}{\left(x\right)}}{2}\right)\right) + \left(i \left(\frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \frac{\operatorname{re}{\left(x\right)}}{2} - \frac{\sqrt{3} \operatorname{im}{\left(x\right)}}{2}\right)$$
=
/ ___ \ / ___ \ |im(x) \/ 3 *re(x)| |im(x) \/ 3 *re(x)| I*|----- + -----------| + I*|----- - -----------| - I*im(x) \ 2 2 / \ 2 2 /
$$i \left(- \frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + i \left(\frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) - i \operatorname{im}{\left(x\right)}$$
product
/ / ___ \ ___ \ / / ___ \ ___ \
|re(x) |im(x) \/ 3 *re(x)| \/ 3 *im(x)| |re(x) |im(x) \/ 3 *re(x)| \/ 3 *im(x)|
(-re(x) - I*im(x))*|----- + I*|----- - -----------| + -----------|*|----- + I*|----- + -----------| - -----------|
\ 2 \ 2 2 / 2 / \ 2 \ 2 2 / 2 /
$$\left(- \operatorname{re}{\left(x\right)} - i \operatorname{im}{\left(x\right)}\right) \left(i \left(- \frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \frac{\operatorname{re}{\left(x\right)}}{2} + \frac{\sqrt{3} \operatorname{im}{\left(x\right)}}{2}\right) \left(i \left(\frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} + \frac{\operatorname{im}{\left(x\right)}}{2}\right) + \frac{\operatorname{re}{\left(x\right)}}{2} - \frac{\sqrt{3} \operatorname{im}{\left(x\right)}}{2}\right)$$
=
3 3 2 2 - re (x) + I*im (x) + 3*im (x)*re(x) - 3*I*re (x)*im(x)
$$- \left(\operatorname{re}{\left(x\right)}\right)^{3} - 3 i \left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{im}{\left(x\right)} + 3 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{2} + i \left(\operatorname{im}{\left(x\right)}\right)^{3}$$
-re(x)^3 + i*im(x)^3 + 3*im(x)^2*re(x) - 3*i*re(x)^2*im(x)