Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation x/4+13=x+1
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Equation 4(x-8)=-5
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Equation 5(x-2)^2=-6x-44
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Equation 2^x=5
- Express {x} in terms of y where:
- 4*x+8*y=7
- -14*x+5*y=3
- 13*x-18*y=-17
- 4*x+18*y=-7
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Identical expressions
- x^ three + nine *x^ two + twenty-four *x+ eight = zero
- x cubed plus 9 multiply by x squared plus 24 multiply by x plus 8 equally 0
- x to the power of three plus nine multiply by x to the power of two plus twenty minus four multiply by x plus eight equally zero
- x3+9*x2+24*x+8=0
- x³+9*x²+24*x+8=0
- x to the power of 3+9*x to the power of 2+24*x+8=0
- x^3+9x^2+24x+8=0
- x3+9x2+24x+8=0
- x^3+9*x^2+24*x+8=O
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Similar expressions
- x^3-9*x^2+24*x+8=0
- x^3+9*x^2-24*x+8=0
- x^3+9*x^2+24*x-8=0
x^3+9*x^2+24*x+8=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
$$p = \frac{b}{a}$$
$$p = 9$$
$$q = \frac{c}{a}$$
$$q = 24$$
$$v = \frac{d}{a}$$
$$v = 8$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -9$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 24$$
$$x_{1} x_{2} x_{3} = 8$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 9$$
$$q = \frac{c}{a}$$
$$q = 24$$
$$v = \frac{d}{a}$$
$$v = 8$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -9$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 24$$
$$x_{1} x_{2} x_{3} = 8$$
Rapid solution
[src]
_____________ / _____________\
3 / ___ | ___ ___ 3 / ___ |
1 \/ 5 + 2*\/ 6 | \/ 3 \/ 3 *\/ 5 + 2*\/ 6 |
x1 = -3 - ------------------ - ---------------- + I*|------------------ - ----------------------|
_____________ 2 | _____________ 2 |
3 / ___ | 3 / ___ |
2*\/ 5 + 2*\/ 6 \2*\/ 5 + 2*\/ 6 /
$$x_{1} = -3 - \frac{\sqrt[3]{2 \sqrt{6} + 5}}{2} - \frac{1}{2 \sqrt[3]{2 \sqrt{6} + 5}} + i \left(- \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}}\right)$$
_____________ / _____________ \
3 / ___ | ___ 3 / ___ ___ |
1 \/ 5 + 2*\/ 6 |\/ 3 *\/ 5 + 2*\/ 6 \/ 3 |
x2 = -3 - ------------------ - ---------------- + I*|---------------------- - ------------------|
_____________ 2 | 2 _____________|
3 / ___ | 3 / ___ |
2*\/ 5 + 2*\/ 6 \ 2*\/ 5 + 2*\/ 6 /
$$x_{2} = -3 - \frac{\sqrt[3]{2 \sqrt{6} + 5}}{2} - \frac{1}{2 \sqrt[3]{2 \sqrt{6} + 5}} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}} + \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2}\right)$$
_____________
1 3 / ___
x3 = -3 + ---------------- + \/ 5 + 2*\/ 6
_____________
3 / ___
\/ 5 + 2*\/ 6
$$x_{3} = -3 + \frac{1}{\sqrt[3]{2 \sqrt{6} + 5}} + \sqrt[3]{2 \sqrt{6} + 5}$$
Sum and product of roots
[src]
sum
_____________ / _____________\ _____________ / _____________ \
3 / ___ | ___ ___ 3 / ___ | 3 / ___ | ___ 3 / ___ ___ | _____________
1 \/ 5 + 2*\/ 6 | \/ 3 \/ 3 *\/ 5 + 2*\/ 6 | 1 \/ 5 + 2*\/ 6 |\/ 3 *\/ 5 + 2*\/ 6 \/ 3 | 1 3 / ___
0 + -3 - ------------------ - ---------------- + I*|------------------ - ----------------------| + -3 - ------------------ - ---------------- + I*|---------------------- - ------------------| + -3 + ---------------- + \/ 5 + 2*\/ 6
_____________ 2 | _____________ 2 | _____________ 2 | 2 _____________| _____________
3 / ___ | 3 / ___ | 3 / ___ | 3 / ___ | 3 / ___
2*\/ 5 + 2*\/ 6 \2*\/ 5 + 2*\/ 6 / 2*\/ 5 + 2*\/ 6 \ 2*\/ 5 + 2*\/ 6 / \/ 5 + 2*\/ 6
$$\left(-3 + \frac{1}{\sqrt[3]{2 \sqrt{6} + 5}} + \sqrt[3]{2 \sqrt{6} + 5}\right) - \left(\frac{1}{\sqrt[3]{2 \sqrt{6} + 5}} + \sqrt[3]{2 \sqrt{6} + 5} + 6 - i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}} + \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2}\right) - i \left(- \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}}\right)\right)$$
=
/ _____________\ / _____________ \
| ___ ___ 3 / ___ | | ___ 3 / ___ ___ |
| \/ 3 \/ 3 *\/ 5 + 2*\/ 6 | |\/ 3 *\/ 5 + 2*\/ 6 \/ 3 |
-9 + I*|------------------ - ----------------------| + I*|---------------------- - ------------------|
| _____________ 2 | | 2 _____________|
| 3 / ___ | | 3 / ___ |
\2*\/ 5 + 2*\/ 6 / \ 2*\/ 5 + 2*\/ 6 /
$$-9 + i \left(- \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}}\right) + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}} + \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2}\right)$$
product
/ _____________ / _____________\\ / _____________ / _____________ \\ | 3 / ___ | ___ ___ 3 / ___ || | 3 / ___ | ___ 3 / ___ ___ || / _____________\ | 1 \/ 5 + 2*\/ 6 | \/ 3 \/ 3 *\/ 5 + 2*\/ 6 || | 1 \/ 5 + 2*\/ 6 |\/ 3 *\/ 5 + 2*\/ 6 \/ 3 || | 1 3 / ___ | 1*|-3 - ------------------ - ---------------- + I*|------------------ - ----------------------||*|-3 - ------------------ - ---------------- + I*|---------------------- - ------------------||*|-3 + ---------------- + \/ 5 + 2*\/ 6 | | _____________ 2 | _____________ 2 || | _____________ 2 | 2 _____________|| | _____________ | | 3 / ___ | 3 / ___ || | 3 / ___ | 3 / ___ || | 3 / ___ | \ 2*\/ 5 + 2*\/ 6 \2*\/ 5 + 2*\/ 6 // \ 2*\/ 5 + 2*\/ 6 \ 2*\/ 5 + 2*\/ 6 // \ \/ 5 + 2*\/ 6 /
$$\left(-3 - \frac{\sqrt[3]{2 \sqrt{6} + 5}}{2} - \frac{1}{2 \sqrt[3]{2 \sqrt{6} + 5}} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}} + \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2}\right)\right) 1 \left(-3 - \frac{\sqrt[3]{2 \sqrt{6} + 5}}{2} - \frac{1}{2 \sqrt[3]{2 \sqrt{6} + 5}} + i \left(- \frac{\sqrt{3} \sqrt[3]{2 \sqrt{6} + 5}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{2 \sqrt{6} + 5}}\right)\right) \left(-3 + \frac{1}{\sqrt[3]{2 \sqrt{6} + 5}} + \sqrt[3]{2 \sqrt{6} + 5}\right)$$
=
-8
$$-8$$
-8
Numerical answer
[src]
x1 = -4.30644393235877 - 1.45615495232862*i
x2 = -4.30644393235877 + 1.45615495232862*i
x3 = -0.387112135282455
x3 = -0.387112135282455
The graph