Mister Exam
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How to use it?
- Equation:
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Equation cos(x)-1=0
-
Equation (x+3)^4-4(x+3)^2-5=0
-
Equation x^2+14*x+24=0
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Equation (x+3)^4+2(x+3)^2-8=0
- Express {x} in terms of y where:
- -16*x-19*y=-4
- -9*x+17*y=-3
- 2*x-20*y=-7
- -1*x+1*y=18
-
Identical expressions
- x^ three -4x^ two +7x- three = zero
- x cubed minus 4x squared plus 7x minus 3 equally 0
- x to the power of three minus 4x to the power of two plus 7x minus three equally zero
- x3-4x2+7x-3=0
- x³-4x²+7x-3=0
- x to the power of 3-4x to the power of 2+7x-3=0
- x^3-4x^2+7x-3=O
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Similar expressions
- x^3+4x^2+7x-3=0
- x^3-4x^2-7x-3=0
- x^3-4x^2+7x+3=0
x^3-4x^2+7x-3=0 equation
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The solution
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[src]
3 2 x - 4*x + 7*x - 3 = 0
$$\left(7 x + \left(x^{3} - 4 x^{2}\right)\right) - 3 = 0$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = 7$$
$$v = \frac{d}{a}$$
$$v = -3$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 4$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 7$$
$$x_{1} x_{2} x_{3} = -3$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = 7$$
$$v = \frac{d}{a}$$
$$v = -3$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 4$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 7$$
$$x_{1} x_{2} x_{3} = -3$$
Sum and product of roots
[src]
sum
_______________ / _______________ \ _______________ / _______________\ _______________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 43 9*\/ 29 | ___ / 43 9*\/ 29 | / 43 9*\/ 29 | ___ / 43 9*\/ 29 | / 43 9*\/ 29
3 / -- + -------- |\/ 3 *3 / -- + -------- ___ | 3 / -- + -------- | ___ \/ 3 *3 / -- + -------- | 3 / -- + --------
4 5 \/ 2 2 | \/ 2 2 5*\/ 3 | 4 5 \/ 2 2 | 5*\/ 3 \/ 2 2 | 4 \/ 2 2 5
- - ---------------------- + -------------------- + I*|-------------------------- + ----------------------| + - - ---------------------- + -------------------- + I*|- ---------------------- - --------------------------| + - - -------------------- + ----------------------
3 _______________ 6 | 6 _______________| 3 _______________ 6 | _______________ 6 | 3 3 _______________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 43 9*\/ 29 | / 43 9*\/ 29 | / 43 9*\/ 29 | / 43 9*\/ 29 | / 43 9*\/ 29
6*3 / -- + -------- | 6*3 / -- + -------- | 6*3 / -- + -------- | 6*3 / -- + -------- | 3*3 / -- + --------
\/ 2 2 \ \/ 2 2 / \/ 2 2 \ \/ 2 2 / \/ 2 2
$$\left(- \frac{\sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{3} + \frac{5}{3 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}} + \frac{4}{3}\right) + \left(\left(- \frac{5}{6 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}} + \frac{\sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{6} + \frac{4}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{6} - \frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}\right)\right) + \left(- \frac{5}{6 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}} + \frac{\sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{6} + \frac{4}{3} + i \left(\frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{6}\right)\right)\right)$$
=
/ _______________\ / _______________ \
| / ____ | | / ____ |
| ___ / 43 9*\/ 29 | | ___ / 43 9*\/ 29 |
| ___ \/ 3 *3 / -- + -------- | |\/ 3 *3 / -- + -------- ___ |
| 5*\/ 3 \/ 2 2 | | \/ 2 2 5*\/ 3 |
4 + I*|- ---------------------- - --------------------------| + I*|-------------------------- + ----------------------|
| _______________ 6 | | 6 _______________|
| / ____ | | / ____ |
| / 43 9*\/ 29 | | / 43 9*\/ 29 |
| 6*3 / -- + -------- | | 6*3 / -- + -------- |
\ \/ 2 2 / \ \/ 2 2 /
$$4 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{6} - \frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}\right) + i \left(\frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{6}\right)$$
product
/ _______________ / _______________ \\ / _______________ / _______________\\ / _______________ \ | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 43 9*\/ 29 | ___ / 43 9*\/ 29 || | / 43 9*\/ 29 | ___ / 43 9*\/ 29 || | / 43 9*\/ 29 | | 3 / -- + -------- |\/ 3 *3 / -- + -------- ___ || | 3 / -- + -------- | ___ \/ 3 *3 / -- + -------- || | 3 / -- + -------- | |4 5 \/ 2 2 | \/ 2 2 5*\/ 3 || |4 5 \/ 2 2 | 5*\/ 3 \/ 2 2 || |4 \/ 2 2 5 | |- - ---------------------- + -------------------- + I*|-------------------------- + ----------------------||*|- - ---------------------- + -------------------- + I*|- ---------------------- - --------------------------||*|- - -------------------- + ----------------------| |3 _______________ 6 | 6 _______________|| |3 _______________ 6 | _______________ 6 || |3 3 _______________| | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 43 9*\/ 29 | / 43 9*\/ 29 || | / 43 9*\/ 29 | / 43 9*\/ 29 || | / 43 9*\/ 29 | | 6*3 / -- + -------- | 6*3 / -- + -------- || | 6*3 / -- + -------- | 6*3 / -- + -------- || | 3*3 / -- + -------- | \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 /
$$\left(- \frac{5}{6 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}} + \frac{\sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{6} + \frac{4}{3} + i \left(\frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{6}\right)\right) \left(- \frac{5}{6 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}} + \frac{\sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{6} + \frac{4}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{6} - \frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{3} + \frac{5}{3 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}} + \frac{4}{3}\right)$$
=
3
$$3$$
3
Rapid solution
[src]
_______________ / _______________ \
/ ____ | / ____ |
/ 43 9*\/ 29 | ___ / 43 9*\/ 29 |
3 / -- + -------- |\/ 3 *3 / -- + -------- ___ |
4 5 \/ 2 2 | \/ 2 2 5*\/ 3 |
x1 = - - ---------------------- + -------------------- + I*|-------------------------- + ----------------------|
3 _______________ 6 | 6 _______________|
/ ____ | / ____ |
/ 43 9*\/ 29 | / 43 9*\/ 29 |
6*3 / -- + -------- | 6*3 / -- + -------- |
\/ 2 2 \ \/ 2 2 /
$$x_{1} = - \frac{5}{6 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}} + \frac{\sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{6} + \frac{4}{3} + i \left(\frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{6}\right)$$
_______________ / _______________\
/ ____ | / ____ |
/ 43 9*\/ 29 | ___ / 43 9*\/ 29 |
3 / -- + -------- | ___ \/ 3 *3 / -- + -------- |
4 5 \/ 2 2 | 5*\/ 3 \/ 2 2 |
x2 = - - ---------------------- + -------------------- + I*|- ---------------------- - --------------------------|
3 _______________ 6 | _______________ 6 |
/ ____ | / ____ |
/ 43 9*\/ 29 | / 43 9*\/ 29 |
6*3 / -- + -------- | 6*3 / -- + -------- |
\/ 2 2 \ \/ 2 2 /
$$x_{2} = - \frac{5}{6 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}} + \frac{\sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{6} + \frac{4}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{6} - \frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}\right)$$
_______________
/ ____
/ 43 9*\/ 29
3 / -- + --------
4 \/ 2 2 5
x3 = - - -------------------- + ----------------------
3 3 _______________
/ ____
/ 43 9*\/ 29
3*3 / -- + --------
\/ 2 2
$$x_{3} = - \frac{\sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}}{3} + \frac{5}{3 \sqrt[3]{\frac{43}{2} + \frac{9 \sqrt{29}}{2}}} + \frac{4}{3}$$
x3 = -(43/2 + 9*sqrt(29)/2)^(1/3)/3 + 5/(3*(43/2 + 9*sqrt(29)/2)^(1/3)) + 4/3
Numerical answer
[src]
x1 = 1.69632339085132 + 1.43594986410996*i
x2 = 1.69632339085132 - 1.43594986410996*i
x3 = 0.607353218297359
x3 = 0.607353218297359