Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation -5x=30
-
Equation 3*x^2=18*x
-
Equation 16*x^2-9=0
-
Equation 0*x=5
- Express {x} in terms of y where:
- 10*x-18*y=-6
- 3*x-12*y=-14
- 16*x+11*y=-14
- -4*x+20*y=-14
-
Identical expressions
- x^ three +18x+ fifteen = zero
- x cubed plus 18x plus 15 equally 0
- x to the power of three plus 18x plus fifteen equally zero
- x3+18x+15=0
- x³+18x+15=0
- x to the power of 3+18x+15=0
- x^3+18x+15=O
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Similar expressions
- x^3-18x+15=0
- x^3+18x-15=0
x^3+18x+15=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 18$$
$$v = \frac{d}{a}$$
$$v = 15$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 18$$
$$x_{1} x_{2} x_{3} = 15$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 18$$
$$v = \frac{d}{a}$$
$$v = 15$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 18$$
$$x_{1} x_{2} x_{3} = 15$$
Sum and product of roots
[src]
sum
2/3 / 6 ___\ 2/3 / 6 ___\
2/3 3 ___ 3 ___ 3 | 5/6 3*\/ 3 | 3 ___ 3 | 5/6 3*\/ 3 |
3 - 2*\/ 3 + \/ 3 - ---- + I*|- 3 - -------| + \/ 3 - ---- + I*|3 + -------|
2 \ 2 / 2 \ 2 /
$$\left(\left(- 2 \sqrt[3]{3} + 3^{\frac{2}{3}}\right) + \left(- \frac{3^{\frac{2}{3}}}{2} + \sqrt[3]{3} + i \left(- 3^{\frac{5}{6}} - \frac{3 \sqrt[6]{3}}{2}\right)\right)\right) + \left(- \frac{3^{\frac{2}{3}}}{2} + \sqrt[3]{3} + i \left(\frac{3 \sqrt[6]{3}}{2} + 3^{\frac{5}{6}}\right)\right)$$
=
/ 6 ___\ / 6 ___\ | 5/6 3*\/ 3 | | 5/6 3*\/ 3 | I*|3 + -------| + I*|- 3 - -------| \ 2 / \ 2 /
$$i \left(- 3^{\frac{5}{6}} - \frac{3 \sqrt[6]{3}}{2}\right) + i \left(\frac{3 \sqrt[6]{3}}{2} + 3^{\frac{5}{6}}\right)$$
product
/ 2/3 / 6 ___\\ / 2/3 / 6 ___\\
/ 2/3 3 ___\ |3 ___ 3 | 5/6 3*\/ 3 || |3 ___ 3 | 5/6 3*\/ 3 ||
\3 - 2*\/ 3 /*|\/ 3 - ---- + I*|- 3 - -------||*|\/ 3 - ---- + I*|3 + -------||
\ 2 \ 2 // \ 2 \ 2 //
$$\left(- 2 \sqrt[3]{3} + 3^{\frac{2}{3}}\right) \left(- \frac{3^{\frac{2}{3}}}{2} + \sqrt[3]{3} + i \left(- 3^{\frac{5}{6}} - \frac{3 \sqrt[6]{3}}{2}\right)\right) \left(- \frac{3^{\frac{2}{3}}}{2} + \sqrt[3]{3} + i \left(\frac{3 \sqrt[6]{3}}{2} + 3^{\frac{5}{6}}\right)\right)$$
=
-15
$$-15$$
-15
Rapid solution
[src]
2/3 3 ___ x1 = 3 - 2*\/ 3
$$x_{1} = - 2 \sqrt[3]{3} + 3^{\frac{2}{3}}$$
2/3 / 6 ___\
3 ___ 3 | 5/6 3*\/ 3 |
x2 = \/ 3 - ---- + I*|- 3 - -------|
2 \ 2 /
$$x_{2} = - \frac{3^{\frac{2}{3}}}{2} + \sqrt[3]{3} + i \left(- 3^{\frac{5}{6}} - \frac{3 \sqrt[6]{3}}{2}\right)$$
2/3 / 6 ___\
3 ___ 3 | 5/6 3*\/ 3 |
x3 = \/ 3 - ---- + I*|3 + -------|
2 \ 2 /
$$x_{3} = - \frac{3^{\frac{2}{3}}}{2} + \sqrt[3]{3} + i \left(\frac{3 \sqrt[6]{3}}{2} + 3^{\frac{5}{6}}\right)$$
x3 = -3^(2/3)/2 + 3^(1/3) + i*(3*3^(1/6)/2 + 3^(5/6))
Numerical answer
[src]
x1 = -0.804415317562913
x2 = 0.402207658781456 + 4.29945496573082*i
x3 = 0.402207658781456 - 4.29945496573082*i
x3 = 0.402207658781456 - 4.29945496573082*i