Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation 5x-9=3x+1
-
Equation -9*(8-9x)=4x+5
-
Equation (x^2+4)/x=0
-
Equation x^2+4x+2=0
- Express {x} in terms of y where:
- sqrt(x)+sqrt(y)=69,49
- (x^2+y^2)×(x+y-3)=2xy
- sqrt(1-x)^2+(4-y)^2=0
- lnx+sin(y/x+2x)=4
-
Identical expressions
- x^ three - five *x- nine = zero
- x cubed minus 5 multiply by x minus 9 equally 0
- x to the power of three minus five multiply by x minus nine equally zero
- x3-5*x-9=0
- x³-5*x-9=0
- x to the power of 3-5*x-9=0
- x^3-5x-9=0
- x3-5x-9=0
- x^3-5*x-9=O
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Similar expressions
- x^3+5*x-9=0
- x^3-5*x+9=0
x^3-5*x-9=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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -5$$
$$v = \frac{d}{a}$$
$$v = -9$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -5$$
$$x_{1} x_{2} x_{3} = -9$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -5$$
$$v = \frac{d}{a}$$
$$v = -9$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -5$$
$$x_{1} x_{2} x_{3} = -9$$
Sum and product of roots
[src]
sum
______________ / ______________ \ ______________ / ______________ \
/ ______ | / ______ | / ______ | / ______ |
/ 9 \/ 5061 | ___ / 9 \/ 5061 | / 9 \/ 5061 | ___ / 9 \/ 5061 | ______________
3 / - + -------- | \/ 3 *3 / - + -------- ___ | 3 / - + -------- |\/ 3 *3 / - + -------- ___ | / ______
5 \/ 2 18 | \/ 2 18 5*\/ 3 | 5 \/ 2 18 | \/ 2 18 5*\/ 3 | / 9 \/ 5061 5
- --------------------- - ------------------- + I*|- ------------------------- + ---------------------| + - --------------------- - ------------------- + I*|------------------------- - ---------------------| + 3 / - + -------- + ---------------------
______________ 2 | 2 ______________| ______________ 2 | 2 ______________| \/ 2 18 ______________
/ ______ | / ______ | / ______ | / ______ | / ______
/ 9 \/ 5061 | / 9 \/ 5061 | / 9 \/ 5061 | / 9 \/ 5061 | / 9 \/ 5061
6*3 / - + -------- | 6*3 / - + -------- | 6*3 / - + -------- | 6*3 / - + -------- | 3*3 / - + --------
\/ 2 18 \ \/ 2 18 / \/ 2 18 \ \/ 2 18 / \/ 2 18
$$\left(\frac{5}{3 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}{2} - \frac{5}{6 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}{2} + \frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}{2} - \frac{5}{6 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}} + i \left(- \frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}{2}\right)\right)\right)$$
=
/ ______________ \ / ______________ \ | / ______ | | / ______ | | ___ / 9 \/ 5061 | | ___ / 9 \/ 5061 | |\/ 3 *3 / - + -------- ___ | | \/ 3 *3 / - + -------- ___ | | \/ 2 18 5*\/ 3 | | \/ 2 18 5*\/ 3 | I*|------------------------- - ---------------------| + I*|- ------------------------- + ---------------------| | 2 ______________| | 2 ______________| | / ______ | | / ______ | | / 9 \/ 5061 | | / 9 \/ 5061 | | 6*3 / - + -------- | | 6*3 / - + -------- | \ \/ 2 18 / \ \/ 2 18 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}{2} + \frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}\right) + i \left(- \frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}{2}\right)$$
product
/ ______________ / ______________ \\ / ______________ / ______________ \\ | / ______ | / ______ || | / ______ | / ______ || | / 9 \/ 5061 | ___ / 9 \/ 5061 || | / 9 \/ 5061 | ___ / 9 \/ 5061 || / ______________ \ | 3 / - + -------- | \/ 3 *3 / - + -------- ___ || | 3 / - + -------- |\/ 3 *3 / - + -------- ___ || | / ______ | | 5 \/ 2 18 | \/ 2 18 5*\/ 3 || | 5 \/ 2 18 | \/ 2 18 5*\/ 3 || | / 9 \/ 5061 5 | |- --------------------- - ------------------- + I*|- ------------------------- + ---------------------||*|- --------------------- - ------------------- + I*|------------------------- - ---------------------||*|3 / - + -------- + ---------------------| | ______________ 2 | 2 ______________|| | ______________ 2 | 2 ______________|| |\/ 2 18 ______________| | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 9 \/ 5061 | / 9 \/ 5061 || | / 9 \/ 5061 | / 9 \/ 5061 || | / 9 \/ 5061 | | 6*3 / - + -------- | 6*3 / - + -------- || | 6*3 / - + -------- | 6*3 / - + -------- || | 3*3 / - + -------- | \ \/ 2 18 \ \/ 2 18 // \ \/ 2 18 \ \/ 2 18 // \ \/ 2 18 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}{2} - \frac{5}{6 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}} + i \left(- \frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}{2} - \frac{5}{6 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}{2} + \frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}\right)\right) \left(\frac{5}{3 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}\right)$$
=
9
$$9$$
9
Rapid solution
[src]
______________ / ______________ \
/ ______ | / ______ |
/ 9 \/ 5061 | ___ / 9 \/ 5061 |
3 / - + -------- | \/ 3 *3 / - + -------- ___ |
5 \/ 2 18 | \/ 2 18 5*\/ 3 |
x1 = - --------------------- - ------------------- + I*|- ------------------------- + ---------------------|
______________ 2 | 2 ______________|
/ ______ | / ______ |
/ 9 \/ 5061 | / 9 \/ 5061 |
6*3 / - + -------- | 6*3 / - + -------- |
\/ 2 18 \ \/ 2 18 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}{2} - \frac{5}{6 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}{2} + \frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}\right)$$
______________ / ______________ \
/ ______ | / ______ |
/ 9 \/ 5061 | ___ / 9 \/ 5061 |
3 / - + -------- |\/ 3 *3 / - + -------- ___ |
5 \/ 2 18 | \/ 2 18 5*\/ 3 |
x2 = - --------------------- - ------------------- + I*|------------------------- - ---------------------|
______________ 2 | 2 ______________|
/ ______ | / ______ |
/ 9 \/ 5061 | / 9 \/ 5061 |
6*3 / - + -------- | 6*3 / - + -------- |
\/ 2 18 \ \/ 2 18 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}{2} - \frac{5}{6 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}} + i \left(- \frac{5 \sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}}{2}\right)$$
______________
/ ______
/ 9 \/ 5061 5
x3 = 3 / - + -------- + ---------------------
\/ 2 18 ______________
/ ______
/ 9 \/ 5061
3*3 / - + --------
\/ 2 18
$$x_{3} = \frac{5}{3 \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{5061}}{18} + \frac{9}{2}}$$
x3 = 5/(3*(sqrt(5061)/18 + 9/2)^(1/3)) + (sqrt(5061)/18 + 9/2)^(1/3)
Numerical answer
[src]
x1 = -1.42759826966035 - 1.05551430999854*i
x2 = 2.8551965393207
x3 = -1.42759826966035 + 1.05551430999854*i
x3 = -1.42759826966035 + 1.05551430999854*i