Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^2-8x+15=0
-
Equation 3x-2=5
-
Equation 3-3*6+2=x
-
Equation sin(3x)=0
- Express {x} in terms of y where:
- -17*x+13*y=-3
- -16*x-2*y=4
- -20*x-13*y=15
- 1*x-6*y=-19
- Graphing y =:
- x^3-3x^2+5
- Integral of d{x}:
- x^3-3x^2+5
-
Identical expressions
- x^ three -3x^ two + five
- x cubed minus 3x squared plus 5
- x to the power of three minus 3x to the power of two plus five
- x3-3x2+5
- x³-3x²+5
- x to the power of 3-3x to the power of 2+5
-
Similar expressions
- x^3+3x^2+5
- x^3-3x^2-5
x^3-3x^2+5 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = -3$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = 5$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 3$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = 5$$
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = -3$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = 5$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 3$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = 5$$
Rapid solution
[src]
_______________ / _______________\
/ ___ | / ___ |
/ 81 27*\/ 5 | ___ / 81 27*\/ 5 |
3 / -- + -------- | ___ \/ 3 *3 / -- + -------- |
\/ 2 2 3 | 3*\/ 3 \/ 2 2 |
x1 = 1 + -------------------- + ---------------------- + I*|- ---------------------- + --------------------------|
6 _______________ | _______________ 6 |
/ ___ | / ___ |
/ 81 27*\/ 5 | / 81 27*\/ 5 |
2*3 / -- + -------- | 2*3 / -- + -------- |
\/ 2 2 \ \/ 2 2 /
$$x_{1} = \frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + 1 + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6}\right)$$
_______________ / _______________ \
/ ___ | / ___ |
/ 81 27*\/ 5 | ___ / 81 27*\/ 5 |
3 / -- + -------- | \/ 3 *3 / -- + -------- ___ |
\/ 2 2 3 | \/ 2 2 3*\/ 3 |
x2 = 1 + -------------------- + ---------------------- + I*|- -------------------------- + ----------------------|
6 _______________ | 6 _______________|
/ ___ | / ___ |
/ 81 27*\/ 5 | / 81 27*\/ 5 |
2*3 / -- + -------- | 2*3 / -- + -------- |
\/ 2 2 \ \/ 2 2 /
$$x_{2} = \frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + 1 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right)$$
_______________
/ ___
/ 81 27*\/ 5
3 / -- + --------
3 \/ 2 2
x3 = 1 - -------------------- - --------------------
_______________ 3
/ ___
/ 81 27*\/ 5
3 / -- + --------
\/ 2 2
$$x_{3} = - \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + 1$$
Sum and product of roots
[src]
sum
_______________ / _______________\ _______________ / _______________ \ _______________
/ ___ | / ___ | / ___ | / ___ | / ___
/ 81 27*\/ 5 | ___ / 81 27*\/ 5 | / 81 27*\/ 5 | ___ / 81 27*\/ 5 | / 81 27*\/ 5
3 / -- + -------- | ___ \/ 3 *3 / -- + -------- | 3 / -- + -------- | \/ 3 *3 / -- + -------- ___ | 3 / -- + --------
\/ 2 2 3 | 3*\/ 3 \/ 2 2 | \/ 2 2 3 | \/ 2 2 3*\/ 3 | 3 \/ 2 2
0 + 1 + -------------------- + ---------------------- + I*|- ---------------------- + --------------------------| + 1 + -------------------- + ---------------------- + I*|- -------------------------- + ----------------------| + 1 - -------------------- - --------------------
6 _______________ | _______________ 6 | 6 _______________ | 6 _______________| _______________ 3
/ ___ | / ___ | / ___ | / ___ | / ___
/ 81 27*\/ 5 | / 81 27*\/ 5 | / 81 27*\/ 5 | / 81 27*\/ 5 | / 81 27*\/ 5
2*3 / -- + -------- | 2*3 / -- + -------- | 2*3 / -- + -------- | 2*3 / -- + -------- | 3 / -- + --------
\/ 2 2 \ \/ 2 2 / \/ 2 2 \ \/ 2 2 / \/ 2 2
$$\left(- \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + 1\right) + \left(\left(\frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + 1 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right)\right) + \left(0 + \left(\frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + 1 + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6}\right)\right)\right)\right)$$
=
/ _______________\ / _______________ \
| / ___ | | / ___ |
| ___ / 81 27*\/ 5 | | ___ / 81 27*\/ 5 |
| ___ \/ 3 *3 / -- + -------- | | \/ 3 *3 / -- + -------- ___ |
| 3*\/ 3 \/ 2 2 | | \/ 2 2 3*\/ 3 |
3 + I*|- ---------------------- + --------------------------| + I*|- -------------------------- + ----------------------|
| _______________ 6 | | 6 _______________|
| / ___ | | / ___ |
| / 81 27*\/ 5 | | / 81 27*\/ 5 |
| 2*3 / -- + -------- | | 2*3 / -- + -------- |
\ \/ 2 2 / \ \/ 2 2 /
$$3 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right) + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6}\right)$$
product
/ _______________ / _______________\\ / _______________ / _______________ \\ / _______________\ | / ___ | / ___ || | / ___ | / ___ || | / ___ | | / 81 27*\/ 5 | ___ / 81 27*\/ 5 || | / 81 27*\/ 5 | ___ / 81 27*\/ 5 || | / 81 27*\/ 5 | | 3 / -- + -------- | ___ \/ 3 *3 / -- + -------- || | 3 / -- + -------- | \/ 3 *3 / -- + -------- ___ || | 3 / -- + -------- | | \/ 2 2 3 | 3*\/ 3 \/ 2 2 || | \/ 2 2 3 | \/ 2 2 3*\/ 3 || | 3 \/ 2 2 | 1*|1 + -------------------- + ---------------------- + I*|- ---------------------- + --------------------------||*|1 + -------------------- + ---------------------- + I*|- -------------------------- + ----------------------||*|1 - -------------------- - --------------------| | 6 _______________ | _______________ 6 || | 6 _______________ | 6 _______________|| | _______________ 3 | | / ___ | / ___ || | / ___ | / ___ || | / ___ | | / 81 27*\/ 5 | / 81 27*\/ 5 || | / 81 27*\/ 5 | / 81 27*\/ 5 || | / 81 27*\/ 5 | | 2*3 / -- + -------- | 2*3 / -- + -------- || | 2*3 / -- + -------- | 2*3 / -- + -------- || | 3 / -- + -------- | \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 /
$$1 \cdot \left(\frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + 1 + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6}\right)\right) \left(\frac{3}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + 1 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27 \sqrt{5}}{2} + \frac{81}{2}}} + 1\right)$$
=
-5
$$-5$$
-5
Numerical answer
[src]
x1 = -1.10380340273554
x2 = 2.05190170136777 - 0.565235851677171*i
x3 = 2.05190170136777 + 0.565235851677171*i
x3 = 2.05190170136777 + 0.565235851677171*i
The graph