Mister Exam
Other calculators
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How to use it?
- Equation:
- Equation (a-2)(b+5)=(a-2)b+(a-2)5
-
Equation x+5=0
-
Equation 4*x^2-25=0
-
Equation x^2-x-7=0
- Express {x} in terms of y where:
- 14*x+3*y=-1
- -8*x-2*y=-11
- -12*x-19*y=-8
- 12*x-14*y=14
-
Identical expressions
- x^ three + twelve *x- twenty = zero
- x cubed plus 12 multiply by x minus 20 equally 0
- x to the power of three plus twelve multiply by x minus twenty equally zero
- x3+12*x-20=0
- x³+12*x-20=0
- x to the power of 3+12*x-20=0
- x^3+12x-20=0
- x3+12x-20=0
- x^3+12*x-20=O
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Similar expressions
- x^3-12*x-20=0
- x^3+12*x+20=0
x^3+12*x-20=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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 12$$
$$v = \frac{d}{a}$$
$$v = -20$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 12$$
$$x_{1} x_{2} x_{3} = -20$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 12$$
$$v = \frac{d}{a}$$
$$v = -20$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 12$$
$$x_{1} x_{2} x_{3} = -20$$
Rapid solution
[src]
_______________ / _______________\
3 / ____ | ___ ___ 3 / ____ |
2 \/ 10 + 2*\/ 41 | 2*\/ 3 \/ 3 *\/ 10 + 2*\/ 41 |
x1 = ------------------ - ------------------ + I*|- ------------------ - ------------------------|
_______________ 2 | _______________ 2 |
3 / ____ | 3 / ____ |
\/ 10 + 2*\/ 41 \ \/ 10 + 2*\/ 41 /
$$x_{1} = - \frac{\sqrt[3]{10 + 2 \sqrt{41}}}{2} + \frac{2}{\sqrt[3]{10 + 2 \sqrt{41}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2} - \frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}}\right)$$
_______________ / _______________ \
3 / ____ | ___ 3 / ____ ___ |
2 \/ 10 + 2*\/ 41 |\/ 3 *\/ 10 + 2*\/ 41 2*\/ 3 |
x2 = ------------------ - ------------------ + I*|------------------------ + ------------------|
_______________ 2 | 2 _______________|
3 / ____ | 3 / ____ |
\/ 10 + 2*\/ 41 \ \/ 10 + 2*\/ 41 /
$$x_{2} = - \frac{\sqrt[3]{10 + 2 \sqrt{41}}}{2} + \frac{2}{\sqrt[3]{10 + 2 \sqrt{41}}} + i \left(\frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}} + \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2}\right)$$
_______________
3 / ____ 4
x3 = \/ 10 + 2*\/ 41 - ------------------
_______________
3 / ____
\/ 10 + 2*\/ 41
$$x_{3} = - \frac{4}{\sqrt[3]{10 + 2 \sqrt{41}}} + \sqrt[3]{10 + 2 \sqrt{41}}$$
x3 = -4/(10 + 2*sqrt(41))^(1/3) + (10 + 2*sqrt(41))^(1/3)
Sum and product of roots
[src]
sum
_______________ / _______________\ _______________ / _______________ \
3 / ____ | ___ ___ 3 / ____ | 3 / ____ | ___ 3 / ____ ___ | _______________
2 \/ 10 + 2*\/ 41 | 2*\/ 3 \/ 3 *\/ 10 + 2*\/ 41 | 2 \/ 10 + 2*\/ 41 |\/ 3 *\/ 10 + 2*\/ 41 2*\/ 3 | 3 / ____ 4
------------------ - ------------------ + I*|- ------------------ - ------------------------| + ------------------ - ------------------ + I*|------------------------ + ------------------| + \/ 10 + 2*\/ 41 - ------------------
_______________ 2 | _______________ 2 | _______________ 2 | 2 _______________| _______________
3 / ____ | 3 / ____ | 3 / ____ | 3 / ____ | 3 / ____
\/ 10 + 2*\/ 41 \ \/ 10 + 2*\/ 41 / \/ 10 + 2*\/ 41 \ \/ 10 + 2*\/ 41 / \/ 10 + 2*\/ 41
$$\left(- \frac{4}{\sqrt[3]{10 + 2 \sqrt{41}}} + \sqrt[3]{10 + 2 \sqrt{41}}\right) + \left(\left(- \frac{\sqrt[3]{10 + 2 \sqrt{41}}}{2} + \frac{2}{\sqrt[3]{10 + 2 \sqrt{41}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2} - \frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}}\right)\right) + \left(- \frac{\sqrt[3]{10 + 2 \sqrt{41}}}{2} + \frac{2}{\sqrt[3]{10 + 2 \sqrt{41}}} + i \left(\frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}} + \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2}\right)\right)\right)$$
=
/ _______________ \ / _______________\ | ___ 3 / ____ ___ | | ___ ___ 3 / ____ | |\/ 3 *\/ 10 + 2*\/ 41 2*\/ 3 | | 2*\/ 3 \/ 3 *\/ 10 + 2*\/ 41 | I*|------------------------ + ------------------| + I*|- ------------------ - ------------------------| | 2 _______________| | _______________ 2 | | 3 / ____ | | 3 / ____ | \ \/ 10 + 2*\/ 41 / \ \/ 10 + 2*\/ 41 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2} - \frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}}\right) + i \left(\frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}} + \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2}\right)$$
product
/ _______________ / _______________\\ / _______________ / _______________ \\ | 3 / ____ | ___ ___ 3 / ____ || | 3 / ____ | ___ 3 / ____ ___ || / _______________ \ | 2 \/ 10 + 2*\/ 41 | 2*\/ 3 \/ 3 *\/ 10 + 2*\/ 41 || | 2 \/ 10 + 2*\/ 41 |\/ 3 *\/ 10 + 2*\/ 41 2*\/ 3 || |3 / ____ 4 | |------------------ - ------------------ + I*|- ------------------ - ------------------------||*|------------------ - ------------------ + I*|------------------------ + ------------------||*|\/ 10 + 2*\/ 41 - ------------------| | _______________ 2 | _______________ 2 || | _______________ 2 | 2 _______________|| | _______________| |3 / ____ | 3 / ____ || |3 / ____ | 3 / ____ || | 3 / ____ | \\/ 10 + 2*\/ 41 \ \/ 10 + 2*\/ 41 // \\/ 10 + 2*\/ 41 \ \/ 10 + 2*\/ 41 // \ \/ 10 + 2*\/ 41 /
$$\left(- \frac{\sqrt[3]{10 + 2 \sqrt{41}}}{2} + \frac{2}{\sqrt[3]{10 + 2 \sqrt{41}}} + i \left(\frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}} + \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2}\right)\right) \left(- \frac{\sqrt[3]{10 + 2 \sqrt{41}}}{2} + \frac{2}{\sqrt[3]{10 + 2 \sqrt{41}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{10 + 2 \sqrt{41}}}{2} - \frac{2 \sqrt{3}}{\sqrt[3]{10 + 2 \sqrt{41}}}\right)\right) \left(- \frac{4}{\sqrt[3]{10 + 2 \sqrt{41}}} + \sqrt[3]{10 + 2 \sqrt{41}}\right)$$
=
20
$$20$$
20
Numerical answer
[src]
x1 = -0.712675737640377 - 3.67746109715164*i
x2 = 1.42535147528075
x3 = -0.712675737640377 + 3.67746109715164*i
x3 = -0.712675737640377 + 3.67746109715164*i