Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation sin(x)=cos(x)
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Equation x^3=3
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Equation cos(x)=(-1/2)
-
Equation sin(x-1)=0
- Express {x} in terms of y where:
- -7*x-20*y=-5
- 5*x-6*y=3
- -20*x+17*y=-16
- -8*x-2*y=-15
-
Identical expressions
- -x^ three - sixty-seven *x+ twelve = zero
- minus x cubed minus 67 multiply by x plus 12 equally 0
- minus x to the power of three minus sixty minus seven multiply by x plus twelve equally zero
- -x3-67*x+12=0
- -x³-67*x+12=0
- -x to the power of 3-67*x+12=0
- -x^3-67x+12=0
- -x3-67x+12=0
- -x^3-67*x+12=O
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Similar expressions
- -x^3-67*x-12=0
- x^3-67*x+12=0
- -x^3+67*x+12=0
-x^3-67*x+12=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Vieta's Theorem
rewrite the equation
$$\left(- x^{3} - 67 x\right) + 12 = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + 67 x - 12 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 67$$
$$v = \frac{d}{a}$$
$$v = -12$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 67$$
$$x_{1} x_{2} x_{3} = -12$$
$$\left(- x^{3} - 67 x\right) + 12 = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + 67 x - 12 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 67$$
$$v = \frac{d}{a}$$
$$v = -12$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 67$$
$$x_{1} x_{2} x_{3} = -12$$
Rapid solution
[src]
________________ / ________________\
/ ________ | / ________ |
/ \/ 905205 | ___ / \/ 905205 |
3 / 6 + ---------- | ___ \/ 3 *3 / 6 + ---------- |
\/ 9 67 | 67*\/ 3 \/ 9 |
x1 = - --------------------- + ----------------------- + I*|- ----------------------- - ---------------------------|
2 ________________ | ________________ 2 |
/ ________ | / ________ |
/ \/ 905205 | / \/ 905205 |
6*3 / 6 + ---------- | 6*3 / 6 + ---------- |
\/ 9 \ \/ 9 /
$$x_{1} = - \frac{\sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}{2} + \frac{67}{6 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}{2} - \frac{67 \sqrt{3}}{6 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}\right)$$
________________ / ________________ \
/ ________ | / ________ |
/ \/ 905205 | ___ / \/ 905205 |
3 / 6 + ---------- |\/ 3 *3 / 6 + ---------- ___ |
\/ 9 67 | \/ 9 67*\/ 3 |
x2 = - --------------------- + ----------------------- + I*|--------------------------- + -----------------------|
2 ________________ | 2 ________________|
/ ________ | / ________ |
/ \/ 905205 | / \/ 905205 |
6*3 / 6 + ---------- | 6*3 / 6 + ---------- |
\/ 9 \ \/ 9 /
$$x_{2} = - \frac{\sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}{2} + \frac{67}{6 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}} + i \left(\frac{67 \sqrt{3}}{6 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}} + \frac{\sqrt{3} \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}{2}\right)$$
________________
/ ________
/ \/ 905205 67
x3 = 3 / 6 + ---------- - -----------------------
\/ 9 ________________
/ ________
/ \/ 905205
3*3 / 6 + ----------
\/ 9
$$x_{3} = - \frac{67}{3 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}} + \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}$$
x3 = -67/(3*(6 + sqrt(905205)/9)^(1/3)) + (6 + sqrt(905205)/9)^(1/3)
Sum and product of roots
[src]
sum
________________ / ________________\ ________________ / ________________ \
/ ________ | / ________ | / ________ | / ________ |
/ \/ 905205 | ___ / \/ 905205 | / \/ 905205 | ___ / \/ 905205 | ________________
3 / 6 + ---------- | ___ \/ 3 *3 / 6 + ---------- | 3 / 6 + ---------- |\/ 3 *3 / 6 + ---------- ___ | / ________
\/ 9 67 | 67*\/ 3 \/ 9 | \/ 9 67 | \/ 9 67*\/ 3 | / \/ 905205 67
- --------------------- + ----------------------- + I*|- ----------------------- - ---------------------------| + - --------------------- + ----------------------- + I*|--------------------------- + -----------------------| + 3 / 6 + ---------- - -----------------------
2 ________________ | ________________ 2 | 2 ________________ | 2 ________________| \/ 9 ________________
/ ________ | / ________ | / ________ | / ________ | / ________
/ \/ 905205 | / \/ 905205 | / \/ 905205 | / \/ 905205 | / \/ 905205
6*3 / 6 + ---------- | 6*3 / 6 + ---------- | 6*3 / 6 + ---------- | 6*3 / 6 + ---------- | 3*3 / 6 + ----------
\/ 9 \ \/ 9 / \/ 9 \ \/ 9 / \/ 9
$$\left(- \frac{67}{3 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}} + \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}\right) + \left(\left(- \frac{\sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}{2} + \frac{67}{6 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}{2} - \frac{67 \sqrt{3}}{6 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}\right)\right) + \left(- \frac{\sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}{2} + \frac{67}{6 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}} + i \left(\frac{67 \sqrt{3}}{6 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}} + \frac{\sqrt{3} \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}{2}\right)\right)\right)$$
=
/ ________________ \ / ________________\ | / ________ | | / ________ | | ___ / \/ 905205 | | ___ / \/ 905205 | |\/ 3 *3 / 6 + ---------- ___ | | ___ \/ 3 *3 / 6 + ---------- | | \/ 9 67*\/ 3 | | 67*\/ 3 \/ 9 | I*|--------------------------- + -----------------------| + I*|- ----------------------- - ---------------------------| | 2 ________________| | ________________ 2 | | / ________ | | / ________ | | / \/ 905205 | | / \/ 905205 | | 6*3 / 6 + ---------- | | 6*3 / 6 + ---------- | \ \/ 9 / \ \/ 9 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}{2} - \frac{67 \sqrt{3}}{6 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}\right) + i \left(\frac{67 \sqrt{3}}{6 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}} + \frac{\sqrt{3} \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}{2}\right)$$
product
/ ________________ / ________________\\ / ________________ / ________________ \\ | / ________ | / ________ || | / ________ | / ________ || | / \/ 905205 | ___ / \/ 905205 || | / \/ 905205 | ___ / \/ 905205 || / ________________ \ | 3 / 6 + ---------- | ___ \/ 3 *3 / 6 + ---------- || | 3 / 6 + ---------- |\/ 3 *3 / 6 + ---------- ___ || | / ________ | | \/ 9 67 | 67*\/ 3 \/ 9 || | \/ 9 67 | \/ 9 67*\/ 3 || | / \/ 905205 67 | |- --------------------- + ----------------------- + I*|- ----------------------- - ---------------------------||*|- --------------------- + ----------------------- + I*|--------------------------- + -----------------------||*|3 / 6 + ---------- - -----------------------| | 2 ________________ | ________________ 2 || | 2 ________________ | 2 ________________|| |\/ 9 ________________| | / ________ | / ________ || | / ________ | / ________ || | / ________ | | / \/ 905205 | / \/ 905205 || | / \/ 905205 | / \/ 905205 || | / \/ 905205 | | 6*3 / 6 + ---------- | 6*3 / 6 + ---------- || | 6*3 / 6 + ---------- | 6*3 / 6 + ---------- || | 3*3 / 6 + ---------- | \ \/ 9 \ \/ 9 // \ \/ 9 \ \/ 9 // \ \/ 9 /
$$\left(- \frac{\sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}{2} + \frac{67}{6 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}} + i \left(\frac{67 \sqrt{3}}{6 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}} + \frac{\sqrt{3} \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}{2}\right)\right) \left(- \frac{\sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}{2} + \frac{67}{6 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}{2} - \frac{67 \sqrt{3}}{6 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}}\right)\right) \left(- \frac{67}{3 \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}} + \sqrt[3]{6 + \frac{\sqrt{905205}}{9}}\right)$$
=
12
$$12$$
12
Numerical answer
[src]
x1 = 0.179018848493067
x2 = -0.0895094242465337 + 8.18682086105998*i
x3 = -0.0895094242465337 - 8.18682086105998*i
x3 = -0.0895094242465337 - 8.18682086105998*i