Mister Exam
Other calculators
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How to use it?
- Equation:
- Equation 1-y^2=2+x
-
Equation (x-5)/2=2*(3x/7-1/2)/3
-
Equation -x=4-3(3x-8)
-
Equation x^2=10
- Express {x} in terms of y where:
- -19*x+1*y=0
- 20*x+13*y=1
- 9*x+17*y=13
- -18*x+12*y=-8
- Derivative of:
- x^3+2x-4
-
Identical expressions
- x^ three +2x- four = zero
- x cubed plus 2x minus 4 equally 0
- x to the power of three plus 2x minus four equally zero
- x3+2x-4=0
- x³+2x-4=0
- x to the power of 3+2x-4=0
- x^3+2x-4=O
-
Similar expressions
- x^3+2x+4=0
- x^3-2x-4=0
x^3+2x-4=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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = -4$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = -4$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = -4$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = -4$$
Rapid solution
[src]
______________ / ______________ \
/ ____ | / ____ |
/ 2*\/ 87 | ___ / 2*\/ 87 |
3 / 2 + -------- | \/ 3 *3 / 2 + -------- ___ |
\/ 9 1 | \/ 9 \/ 3 |
x1 = - ------------------- + --------------------- + I*|- ------------------------- - ---------------------|
2 ______________ | 2 ______________|
/ ____ | / ____ |
/ 2*\/ 87 | / 2*\/ 87 |
3*3 / 2 + -------- | 3*3 / 2 + -------- |
\/ 9 \ \/ 9 /
$$x_{1} = - \frac{\sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} + \frac{1}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} - \frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}\right)$$
______________ / ______________ \
/ ____ | / ____ |
/ 2*\/ 87 | ___ / 2*\/ 87 |
3 / 2 + -------- |\/ 3 *3 / 2 + -------- ___ |
\/ 9 1 | \/ 9 \/ 3 |
x2 = - ------------------- + --------------------- + I*|------------------------- + ---------------------|
2 ______________ | 2 ______________|
/ ____ | / ____ |
/ 2*\/ 87 | / 2*\/ 87 |
3*3 / 2 + -------- | 3*3 / 2 + -------- |
\/ 9 \ \/ 9 /
$$x_{2} = - \frac{\sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} + \frac{1}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2}\right)$$
______________
/ ____
/ 2*\/ 87 2
x3 = 3 / 2 + -------- - ---------------------
\/ 9 ______________
/ ____
/ 2*\/ 87
3*3 / 2 + --------
\/ 9
$$x_{3} = - \frac{2}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}$$
x3 = -2/(3*(2 + 2*sqrt(87)/9)^(1/3)) + (2 + 2*sqrt(87)/9)^(1/3)
Sum and product of roots
[src]
sum
______________ / ______________ \ ______________ / ______________ \
/ ____ | / ____ | / ____ | / ____ |
/ 2*\/ 87 | ___ / 2*\/ 87 | / 2*\/ 87 | ___ / 2*\/ 87 | ______________
3 / 2 + -------- | \/ 3 *3 / 2 + -------- ___ | 3 / 2 + -------- |\/ 3 *3 / 2 + -------- ___ | / ____
\/ 9 1 | \/ 9 \/ 3 | \/ 9 1 | \/ 9 \/ 3 | / 2*\/ 87 2
- ------------------- + --------------------- + I*|- ------------------------- - ---------------------| + - ------------------- + --------------------- + I*|------------------------- + ---------------------| + 3 / 2 + -------- - ---------------------
2 ______________ | 2 ______________| 2 ______________ | 2 ______________| \/ 9 ______________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 2*\/ 87 | / 2*\/ 87 | / 2*\/ 87 | / 2*\/ 87 | / 2*\/ 87
3*3 / 2 + -------- | 3*3 / 2 + -------- | 3*3 / 2 + -------- | 3*3 / 2 + -------- | 3*3 / 2 + --------
\/ 9 \ \/ 9 / \/ 9 \ \/ 9 / \/ 9
$$\left(- \frac{2}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}\right) + \left(\left(- \frac{\sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} + \frac{1}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} - \frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}\right)\right) + \left(- \frac{\sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} + \frac{1}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2}\right)\right)\right)$$
=
/ ______________ \ / ______________ \ | / ____ | | / ____ | | ___ / 2*\/ 87 | | ___ / 2*\/ 87 | |\/ 3 *3 / 2 + -------- ___ | | \/ 3 *3 / 2 + -------- ___ | | \/ 9 \/ 3 | | \/ 9 \/ 3 | I*|------------------------- + ---------------------| + I*|- ------------------------- - ---------------------| | 2 ______________| | 2 ______________| | / ____ | | / ____ | | / 2*\/ 87 | | / 2*\/ 87 | | 3*3 / 2 + -------- | | 3*3 / 2 + -------- | \ \/ 9 / \ \/ 9 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} - \frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}\right) + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2}\right)$$
product
/ ______________ / ______________ \\ / ______________ / ______________ \\ | / ____ | / ____ || | / ____ | / ____ || | / 2*\/ 87 | ___ / 2*\/ 87 || | / 2*\/ 87 | ___ / 2*\/ 87 || / ______________ \ | 3 / 2 + -------- | \/ 3 *3 / 2 + -------- ___ || | 3 / 2 + -------- |\/ 3 *3 / 2 + -------- ___ || | / ____ | | \/ 9 1 | \/ 9 \/ 3 || | \/ 9 1 | \/ 9 \/ 3 || | / 2*\/ 87 2 | |- ------------------- + --------------------- + I*|- ------------------------- - ---------------------||*|- ------------------- + --------------------- + I*|------------------------- + ---------------------||*|3 / 2 + -------- - ---------------------| | 2 ______________ | 2 ______________|| | 2 ______________ | 2 ______________|| |\/ 9 ______________| | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 2*\/ 87 | / 2*\/ 87 || | / 2*\/ 87 | / 2*\/ 87 || | / 2*\/ 87 | | 3*3 / 2 + -------- | 3*3 / 2 + -------- || | 3*3 / 2 + -------- | 3*3 / 2 + -------- || | 3*3 / 2 + -------- | \ \/ 9 \ \/ 9 // \ \/ 9 \ \/ 9 // \ \/ 9 /
$$\left(- \frac{\sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} + \frac{1}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2}\right)\right) \left(- \frac{\sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} + \frac{1}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}{2} - \frac{\sqrt{3}}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}}\right)\right) \left(- \frac{2}{3 \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}} + \sqrt[3]{2 + \frac{2 \sqrt{87}}{9}}\right)$$
=
4
$$4$$
4
Numerical answer
[src]
x1 = -0.589754512301458 - 1.74454325092266*i
x2 = 1.17950902460292
x3 = -0.589754512301458 + 1.74454325092266*i
x3 = -0.589754512301458 + 1.74454325092266*i
The graph