Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation cos(x)-1=0
-
Equation x+5=7
-
Equation 16*x^3+8*x^2+x=0
-
Equation 3/5*y=29/10-1/5
- Express {x} in terms of y where:
- 19*x+2*y=17
- 8*x-9*y=-1
- 13*x-1*y=-20
- 9*x+8*y=-4
- Graphing y =:
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x^4-3x^2+4
- Canonical form:
- x^4-3x^2+4
-
Identical expressions
- x^ four -3x^ two + four = zero
- x to the power of 4 minus 3x squared plus 4 equally 0
- x to the power of four minus 3x to the power of two plus four equally zero
- x4-3x2+4=0
- x⁴-3x²+4=0
- x to the power of 4-3x to the power of 2+4=0
- x^4-3x^2+4=O
-
Similar expressions
- x^4+3x^2+4=0
- x^4-3x^2-4=0
x^4-3x^2+4=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(x^{4} - 3 x^{2}\right) + 4 = 0$$
Do replacement
$$v = x^{2}$$
then the equation will be the:
$$v^{2} - 3 v + 4 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$v_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$v_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -3$$
$$c = 4$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$v_{1} = \frac{3}{2} + \frac{\sqrt{7} i}{2}$$
$$v_{2} = \frac{3}{2} - \frac{\sqrt{7} i}{2}$$
The final answer:
Because
$$v = x^{2}$$
then
$$x_{1} = \sqrt{v_{1}}$$
$$x_{2} = - \sqrt{v_{1}}$$
$$x_{3} = \sqrt{v_{2}}$$
$$x_{4} = - \sqrt{v_{2}}$$
then:
$$x_{1} = $$
$$\frac{0}{1} + \frac{\left(\frac{3}{2} + \frac{\sqrt{7} i}{2}\right)^{\frac{1}{2}}}{1} = \sqrt{\frac{3}{2} + \frac{\sqrt{7} i}{2}}$$
$$x_{2} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(\frac{3}{2} + \frac{\sqrt{7} i}{2}\right)^{\frac{1}{2}}}{1} = - \sqrt{\frac{3}{2} + \frac{\sqrt{7} i}{2}}$$
$$x_{3} = $$
$$\frac{0}{1} + \frac{\left(\frac{3}{2} - \frac{\sqrt{7} i}{2}\right)^{\frac{1}{2}}}{1} = \sqrt{\frac{3}{2} - \frac{\sqrt{7} i}{2}}$$
$$x_{4} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(\frac{3}{2} - \frac{\sqrt{7} i}{2}\right)^{\frac{1}{2}}}{1} = - \sqrt{\frac{3}{2} - \frac{\sqrt{7} i}{2}}$$
$$\left(x^{4} - 3 x^{2}\right) + 4 = 0$$
Do replacement
$$v = x^{2}$$
then the equation will be the:
$$v^{2} - 3 v + 4 = 0$$
This equation is of the form
a*v^2 + b*v + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$v_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$v_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -3$$
$$c = 4$$
, then
D = b^2 - 4 * a * c =
(-3)^2 - 4 * (1) * (4) = -7
Because D<0, then the equation
has no real roots,
but complex roots is exists.
v1 = (-b + sqrt(D)) / (2*a)
v2 = (-b - sqrt(D)) / (2*a)
or
$$v_{1} = \frac{3}{2} + \frac{\sqrt{7} i}{2}$$
$$v_{2} = \frac{3}{2} - \frac{\sqrt{7} i}{2}$$
The final answer:
Because
$$v = x^{2}$$
then
$$x_{1} = \sqrt{v_{1}}$$
$$x_{2} = - \sqrt{v_{1}}$$
$$x_{3} = \sqrt{v_{2}}$$
$$x_{4} = - \sqrt{v_{2}}$$
then:
$$x_{1} = $$
$$\frac{0}{1} + \frac{\left(\frac{3}{2} + \frac{\sqrt{7} i}{2}\right)^{\frac{1}{2}}}{1} = \sqrt{\frac{3}{2} + \frac{\sqrt{7} i}{2}}$$
$$x_{2} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(\frac{3}{2} + \frac{\sqrt{7} i}{2}\right)^{\frac{1}{2}}}{1} = - \sqrt{\frac{3}{2} + \frac{\sqrt{7} i}{2}}$$
$$x_{3} = $$
$$\frac{0}{1} + \frac{\left(\frac{3}{2} - \frac{\sqrt{7} i}{2}\right)^{\frac{1}{2}}}{1} = \sqrt{\frac{3}{2} - \frac{\sqrt{7} i}{2}}$$
$$x_{4} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(\frac{3}{2} - \frac{\sqrt{7} i}{2}\right)^{\frac{1}{2}}}{1} = - \sqrt{\frac{3}{2} - \frac{\sqrt{7} i}{2}}$$
Rapid solution
[src]
/ / ___\\ / / ___\\
| |\/ 7 || | |\/ 7 ||
|atan|-----|| |atan|-----||
___ | \ 3 /| ___ | \ 3 /|
x1 = - \/ 2 *cos|-----------| - I*\/ 2 *sin|-----------|
\ 2 / \ 2 /
$$x_{1} = - \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}$$
/ / ___\\ / / ___\\
| |\/ 7 || | |\/ 7 ||
|atan|-----|| |atan|-----||
___ | \ 3 /| ___ | \ 3 /|
x2 = - \/ 2 *cos|-----------| + I*\/ 2 *sin|-----------|
\ 2 / \ 2 /
$$x_{2} = - \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}$$
/ / ___\\ / / ___\\
| |\/ 7 || | |\/ 7 ||
|atan|-----|| |atan|-----||
___ | \ 3 /| ___ | \ 3 /|
x3 = \/ 2 *cos|-----------| - I*\/ 2 *sin|-----------|
\ 2 / \ 2 /
$$x_{3} = \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}$$
/ / ___\\ / / ___\\
| |\/ 7 || | |\/ 7 ||
|atan|-----|| |atan|-----||
___ | \ 3 /| ___ | \ 3 /|
x4 = \/ 2 *cos|-----------| + I*\/ 2 *sin|-----------|
\ 2 / \ 2 /
$$x_{4} = \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}$$
x4 = sqrt(2)*cos(atan(sqrt(7)/3)/2) + sqrt(2)*i*sin(atan(sqrt(7)/3)/2)
Sum and product of roots
[src]
sum
/ / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\
| |\/ 7 || | |\/ 7 || | |\/ 7 || | |\/ 7 || | |\/ 7 || | |\/ 7 || | |\/ 7 || | |\/ 7 ||
|atan|-----|| |atan|-----|| |atan|-----|| |atan|-----|| |atan|-----|| |atan|-----|| |atan|-----|| |atan|-----||
___ | \ 3 /| ___ | \ 3 /| ___ | \ 3 /| ___ | \ 3 /| ___ | \ 3 /| ___ | \ 3 /| ___ | \ 3 /| ___ | \ 3 /|
- \/ 2 *cos|-----------| - I*\/ 2 *sin|-----------| + - \/ 2 *cos|-----------| + I*\/ 2 *sin|-----------| + \/ 2 *cos|-----------| - I*\/ 2 *sin|-----------| + \/ 2 *cos|-----------| + I*\/ 2 *sin|-----------|
\ 2 / \ 2 / \ 2 / \ 2 / \ 2 / \ 2 / \ 2 / \ 2 /
$$\left(\left(\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right) + \left(\left(- \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right) + \left(- \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right)\right)\right) + \left(\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right)$$
=
0
$$0$$
product
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ | | |\/ 7 || | |\/ 7 ||| | | |\/ 7 || | |\/ 7 ||| | | |\/ 7 || | |\/ 7 ||| | | |\/ 7 || | |\/ 7 ||| | |atan|-----|| |atan|-----||| | |atan|-----|| |atan|-----||| | |atan|-----|| |atan|-----||| | |atan|-----|| |atan|-----||| | ___ | \ 3 /| ___ | \ 3 /|| | ___ | \ 3 /| ___ | \ 3 /|| | ___ | \ 3 /| ___ | \ 3 /|| | ___ | \ 3 /| ___ | \ 3 /|| |- \/ 2 *cos|-----------| - I*\/ 2 *sin|-----------||*|- \/ 2 *cos|-----------| + I*\/ 2 *sin|-----------||*|\/ 2 *cos|-----------| - I*\/ 2 *sin|-----------||*|\/ 2 *cos|-----------| + I*\/ 2 *sin|-----------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(- \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right) \left(- \sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right) \left(\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} - \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right) \left(\sqrt{2} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)} + \sqrt{2} i \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{2} \right)}\right)$$
=
/ ___\
|\/ 7 |
/ ___\ I*atan|-----|
|3 I*\/ 7 | \ 3 /
4*|- - -------|*e
\4 4 /
$$4 \left(\frac{3}{4} - \frac{\sqrt{7} i}{4}\right) e^{i \operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}$$
4*(3/4 - i*sqrt(7)/4)*exp(i*atan(sqrt(7)/3))
Numerical answer
[src]
x1 = -1.3228756555323 + 0.5*i
x2 = 1.3228756555323 - 0.5*i
x3 = 1.3228756555323 + 0.5*i
x4 = -1.3228756555323 - 0.5*i
x4 = -1.3228756555323 - 0.5*i
The graph