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- Equation:
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Equation x^2+4*x+4=0
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Equation (x^2-4)^2+(x^2-3x-10)^2=0
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Equation -x/3+x+7=3
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Equation -20*(x-13)=-220
- Express {x} in terms of y where:
- -14*x+4*y=-16
- -13*x-10*y=5
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Identical expressions
- x^ three -x^ two + ten *x- eight = zero
- x cubed minus x squared plus 10 multiply by x minus 8 equally 0
- x to the power of three minus x to the power of two plus ten multiply by x minus eight equally zero
- x3-x2+10*x-8=0
- x³-x²+10*x-8=0
- x to the power of 3-x to the power of 2+10*x-8=0
- x^3-x^2+10x-8=0
- x3-x2+10x-8=0
- x^3-x^2+10*x-8=O
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Similar expressions
- x^3-x^2-10*x-8=0
- x^3-x^2+10*x+8=0
- x^3+x^2+10*x-8=0
x^3-x^2+10*x-8=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
3 2 x - x + 10*x - 8 = 0
$$\left(10 x + \left(x^{3} - x^{2}\right)\right) - 8 = 0$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 10$$
$$v = \frac{d}{a}$$
$$v = -8$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 10$$
$$x_{1} x_{2} x_{3} = -8$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 10$$
$$v = \frac{d}{a}$$
$$v = -8$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 10$$
$$x_{1} x_{2} x_{3} = -8$$
Sum and product of roots
[src]
sum
_______________ / _______________\ _______________ / _______________ \
/ ______ | / ______ | / ______ | / ______ |
/ 64 \/ 3165 | ___ / 64 \/ 3165 | / 64 \/ 3165 | ___ / 64 \/ 3165 | _______________
3 / -- + -------- | ___ \/ 3 *3 / -- + -------- | 3 / -- + -------- |\/ 3 *3 / -- + -------- ___ | / ______
1 \/ 27 9 29 | 29*\/ 3 \/ 27 9 | 1 \/ 27 9 29 | \/ 27 9 29*\/ 3 | 1 / 64 \/ 3165 29
- - -------------------- + ----------------------- + I*|- ----------------------- - --------------------------| + - - -------------------- + ----------------------- + I*|-------------------------- + -----------------------| + - + 3 / -- + -------- - ----------------------
3 2 _______________ | _______________ 2 | 3 2 _______________ | 2 _______________| 3 \/ 27 9 _______________
/ ______ | / ______ | / ______ | / ______ | / ______
/ 64 \/ 3165 | / 64 \/ 3165 | / 64 \/ 3165 | / 64 \/ 3165 | / 64 \/ 3165
18*3 / -- + -------- | 18*3 / -- + -------- | 18*3 / -- + -------- | 18*3 / -- + -------- | 9*3 / -- + --------
\/ 27 9 \ \/ 27 9 / \/ 27 9 \ \/ 27 9 / \/ 27 9
$$\left(- \frac{29}{9 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}} + \frac{1}{3} + \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}{2} + \frac{1}{3} + \frac{29}{18 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}{2} - \frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}{2} + \frac{1}{3} + \frac{29}{18 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}} + i \left(\frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}} + \frac{\sqrt{3} \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}{2}\right)\right)\right)$$
=
/ _______________ \ / _______________\
| / ______ | | / ______ |
| ___ / 64 \/ 3165 | | ___ / 64 \/ 3165 |
|\/ 3 *3 / -- + -------- ___ | | ___ \/ 3 *3 / -- + -------- |
| \/ 27 9 29*\/ 3 | | 29*\/ 3 \/ 27 9 |
1 + I*|-------------------------- + -----------------------| + I*|- ----------------------- - --------------------------|
| 2 _______________| | _______________ 2 |
| / ______ | | / ______ |
| / 64 \/ 3165 | | / 64 \/ 3165 |
| 18*3 / -- + -------- | | 18*3 / -- + -------- |
\ \/ 27 9 / \ \/ 27 9 /
$$1 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}{2} - \frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}\right) + i \left(\frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}} + \frac{\sqrt{3} \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}{2}\right)$$
product
/ _______________ / _______________\\ / _______________ / _______________ \\ | / ______ | / ______ || | / ______ | / ______ || | / 64 \/ 3165 | ___ / 64 \/ 3165 || | / 64 \/ 3165 | ___ / 64 \/ 3165 || / _______________ \ | 3 / -- + -------- | ___ \/ 3 *3 / -- + -------- || | 3 / -- + -------- |\/ 3 *3 / -- + -------- ___ || | / ______ | |1 \/ 27 9 29 | 29*\/ 3 \/ 27 9 || |1 \/ 27 9 29 | \/ 27 9 29*\/ 3 || |1 / 64 \/ 3165 29 | |- - -------------------- + ----------------------- + I*|- ----------------------- - --------------------------||*|- - -------------------- + ----------------------- + I*|-------------------------- + -----------------------||*|- + 3 / -- + -------- - ----------------------| |3 2 _______________ | _______________ 2 || |3 2 _______________ | 2 _______________|| |3 \/ 27 9 _______________| | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 64 \/ 3165 | / 64 \/ 3165 || | / 64 \/ 3165 | / 64 \/ 3165 || | / 64 \/ 3165 | | 18*3 / -- + -------- | 18*3 / -- + -------- || | 18*3 / -- + -------- | 18*3 / -- + -------- || | 9*3 / -- + -------- | \ \/ 27 9 \ \/ 27 9 // \ \/ 27 9 \ \/ 27 9 // \ \/ 27 9 /
$$\left(- \frac{\sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}{2} + \frac{1}{3} + \frac{29}{18 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}} + i \left(\frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}} + \frac{\sqrt{3} \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}{2} + \frac{1}{3} + \frac{29}{18 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}{2} - \frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}\right)\right) \left(- \frac{29}{9 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}} + \frac{1}{3} + \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}\right)$$
=
8
$$8$$
8
Rapid solution
[src]
_______________ / _______________\
/ ______ | / ______ |
/ 64 \/ 3165 | ___ / 64 \/ 3165 |
3 / -- + -------- | ___ \/ 3 *3 / -- + -------- |
1 \/ 27 9 29 | 29*\/ 3 \/ 27 9 |
x1 = - - -------------------- + ----------------------- + I*|- ----------------------- - --------------------------|
3 2 _______________ | _______________ 2 |
/ ______ | / ______ |
/ 64 \/ 3165 | / 64 \/ 3165 |
18*3 / -- + -------- | 18*3 / -- + -------- |
\/ 27 9 \ \/ 27 9 /
$$x_{1} = - \frac{\sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}{2} + \frac{1}{3} + \frac{29}{18 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}{2} - \frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}\right)$$
_______________ / _______________ \
/ ______ | / ______ |
/ 64 \/ 3165 | ___ / 64 \/ 3165 |
3 / -- + -------- |\/ 3 *3 / -- + -------- ___ |
1 \/ 27 9 29 | \/ 27 9 29*\/ 3 |
x2 = - - -------------------- + ----------------------- + I*|-------------------------- + -----------------------|
3 2 _______________ | 2 _______________|
/ ______ | / ______ |
/ 64 \/ 3165 | / 64 \/ 3165 |
18*3 / -- + -------- | 18*3 / -- + -------- |
\/ 27 9 \ \/ 27 9 /
$$x_{2} = - \frac{\sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}{2} + \frac{1}{3} + \frac{29}{18 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}} + i \left(\frac{29 \sqrt{3}}{18 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}} + \frac{\sqrt{3} \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}}{2}\right)$$
_______________
/ ______
1 / 64 \/ 3165 29
x3 = - + 3 / -- + -------- - ----------------------
3 \/ 27 9 _______________
/ ______
/ 64 \/ 3165
9*3 / -- + --------
\/ 27 9
$$x_{3} = - \frac{29}{9 \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}} + \frac{1}{3} + \sqrt[3]{\frac{64}{27} + \frac{\sqrt{3165}}{9}}$$
x3 = -29/(9*(64/27 + sqrt(3165)/9)^(1/3)) + 1/3 + (64/27 + sqrt(3165)/9)^(1/3)
Numerical answer
[src]
x1 = 0.0938089409870553 - 3.13668335511464*i
x2 = 0.0938089409870553 + 3.13668335511464*i
x3 = 0.812382118025889
x3 = 0.812382118025889