Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation (x-3)*(x+4)=0
-
Equation x^3-x^2-5=0
-
Equation x-2(3x+2)=16
- Equation x^2-1=1+(1/2)*y
- Express {x} in terms of y where:
- -10*x+16*y=-6
- -13*x+9*y=1
- -10*x+9*y=-6
- -17*x+13*y=-3
-
Identical expressions
- x^ three -x^ two - five = zero
- x cubed minus x squared minus 5 equally 0
- x to the power of three minus x to the power of two minus five equally zero
- x3-x2-5=0
- x³-x²-5=0
- x to the power of 3-x to the power of 2-5=0
- x^3-x^2-5=O
-
Similar expressions
- x^3-x^2+5=0
- x^3+x^2-5=0
x^3-x^2-5=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
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -5$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = -5$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -5$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = -5$$
Sum and product of roots
[src]
sum
________________ / ________________ \ ________________ / ________________ \
/ ______ | / ______ | / ______ | / ______ |
/ 137 \/ 2085 | ___ / 137 \/ 2085 | / 137 \/ 2085 | ___ / 137 \/ 2085 | ________________
3 / --- + -------- | \/ 3 *3 / --- + -------- ___ | 3 / --- + -------- |\/ 3 *3 / --- + -------- ___ | / ______
1 \/ 54 18 1 | \/ 54 18 \/ 3 | 1 \/ 54 18 1 | \/ 54 18 \/ 3 | 1 / 137 \/ 2085 1
- - --------------------- - ------------------------ + I*|- --------------------------- + ------------------------| + - - --------------------- - ------------------------ + I*|--------------------------- - ------------------------| + - + 3 / --- + -------- + -----------------------
3 2 ________________ | 2 ________________| 3 2 ________________ | 2 ________________| 3 \/ 54 18 ________________
/ ______ | / ______ | / ______ | / ______ | / ______
/ 137 \/ 2085 | / 137 \/ 2085 | / 137 \/ 2085 | / 137 \/ 2085 | / 137 \/ 2085
18*3 / --- + -------- | 18*3 / --- + -------- | 18*3 / --- + -------- | 18*3 / --- + -------- | 9*3 / --- + --------
\/ 54 18 \ \/ 54 18 / \/ 54 18 \ \/ 54 18 / \/ 54 18
$$\left(\frac{1}{9 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} + \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2}\right)\right)\right)$$
=
/ ________________ \ / ________________ \
| / ______ | | / ______ |
| ___ / 137 \/ 2085 | | ___ / 137 \/ 2085 |
|\/ 3 *3 / --- + -------- ___ | | \/ 3 *3 / --- + -------- ___ |
| \/ 54 18 \/ 3 | | \/ 54 18 \/ 3 |
1 + I*|--------------------------- - ------------------------| + I*|- --------------------------- + ------------------------|
| 2 ________________| | 2 ________________|
| / ______ | | / ______ |
| / 137 \/ 2085 | | / 137 \/ 2085 |
| 18*3 / --- + -------- | | 18*3 / --- + -------- |
\ \/ 54 18 / \ \/ 54 18 /
$$1 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} + \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}\right) + i \left(- \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2}\right)$$
product
/ ________________ / ________________ \\ / ________________ / ________________ \\ | / ______ | / ______ || | / ______ | / ______ || | / 137 \/ 2085 | ___ / 137 \/ 2085 || | / 137 \/ 2085 | ___ / 137 \/ 2085 || / ________________ \ | 3 / --- + -------- | \/ 3 *3 / --- + -------- ___ || | 3 / --- + -------- |\/ 3 *3 / --- + -------- ___ || | / ______ | |1 \/ 54 18 1 | \/ 54 18 \/ 3 || |1 \/ 54 18 1 | \/ 54 18 \/ 3 || |1 / 137 \/ 2085 1 | |- - --------------------- - ------------------------ + I*|- --------------------------- + ------------------------||*|- - --------------------- - ------------------------ + I*|--------------------------- - ------------------------||*|- + 3 / --- + -------- + -----------------------| |3 2 ________________ | 2 ________________|| |3 2 ________________ | 2 ________________|| |3 \/ 54 18 ________________| | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 137 \/ 2085 | / 137 \/ 2085 || | / 137 \/ 2085 | / 137 \/ 2085 || | / 137 \/ 2085 | | 18*3 / --- + -------- | 18*3 / --- + -------- || | 18*3 / --- + -------- | 18*3 / --- + -------- || | 9*3 / --- + -------- | \ \/ 54 18 \ \/ 54 18 // \ \/ 54 18 \ \/ 54 18 // \ \/ 54 18 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} + \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}\right)\right) \left(\frac{1}{9 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}\right)$$
=
5
$$5$$
5
Rapid solution
[src]
________________ / ________________ \
/ ______ | / ______ |
/ 137 \/ 2085 | ___ / 137 \/ 2085 |
3 / --- + -------- | \/ 3 *3 / --- + -------- ___ |
1 \/ 54 18 1 | \/ 54 18 \/ 3 |
x1 = - - --------------------- - ------------------------ + I*|- --------------------------- + ------------------------|
3 2 ________________ | 2 ________________|
/ ______ | / ______ |
/ 137 \/ 2085 | / 137 \/ 2085 |
18*3 / --- + -------- | 18*3 / --- + -------- |
\/ 54 18 \ \/ 54 18 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} + \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}\right)$$
________________ / ________________ \
/ ______ | / ______ |
/ 137 \/ 2085 | ___ / 137 \/ 2085 |
3 / --- + -------- |\/ 3 *3 / --- + -------- ___ |
1 \/ 54 18 1 | \/ 54 18 \/ 3 |
x2 = - - --------------------- - ------------------------ + I*|--------------------------- - ------------------------|
3 2 ________________ | 2 ________________|
/ ______ | / ______ |
/ 137 \/ 2085 | / 137 \/ 2085 |
18*3 / --- + -------- | 18*3 / --- + -------- |
\/ 54 18 \ \/ 54 18 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2}\right)$$
________________
/ ______
1 / 137 \/ 2085 1
x3 = - + 3 / --- + -------- + -----------------------
3 \/ 54 18 ________________
/ ______
/ 137 \/ 2085
9*3 / --- + --------
\/ 54 18
$$x_{3} = \frac{1}{9 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}$$
x3 = 1/(9*(sqrt(2085)/18 + 137/54)^(1/3)) + 1/3 + (sqrt(2085)/18 + 137/54)^(1/3)
Numerical answer
[src]
x1 = 2.11634329862421
x2 = -0.558171649312106 - 1.43213479425353*i
x3 = -0.558171649312106 + 1.43213479425353*i
x3 = -0.558171649312106 + 1.43213479425353*i
The graph