Mister Exam
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Similar expressions
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8x^4-2x^2+1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(8 x^{4} - 2 x^{2}\right) + 1 = 0$$
Do replacement
$$v = x^{2}$$
then the equation will be the:
$$8 v^{2} - 2 v + 1 = 0$$
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
$$a = 8$$
$$b = -2$$
$$c = 1$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$v_{1} = \frac{1}{8} + \frac{\sqrt{7} i}{8}$$
$$v_{2} = \frac{1}{8} - \frac{\sqrt{7} i}{8}$$
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:
$$x_{1} = $$
$$\frac{0}{1} + \frac{\left(\frac{1}{8} + \frac{\sqrt{7} i}{8}\right)^{\frac{1}{2}}}{1} = \sqrt{\frac{1}{8} + \frac{\sqrt{7} i}{8}}$$
$$x_{2} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(\frac{1}{8} + \frac{\sqrt{7} i}{8}\right)^{\frac{1}{2}}}{1} = - \sqrt{\frac{1}{8} + \frac{\sqrt{7} i}{8}}$$
$$x_{3} = $$
$$\frac{0}{1} + \frac{\left(\frac{1}{8} - \frac{\sqrt{7} i}{8}\right)^{\frac{1}{2}}}{1} = \sqrt{\frac{1}{8} - \frac{\sqrt{7} i}{8}}$$
$$x_{4} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(\frac{1}{8} - \frac{\sqrt{7} i}{8}\right)^{\frac{1}{2}}}{1} = - \sqrt{\frac{1}{8} - \frac{\sqrt{7} i}{8}}$$
$$\left(8 x^{4} - 2 x^{2}\right) + 1 = 0$$
Do replacement
$$v = x^{2}$$
then the equation will be the:
$$8 v^{2} - 2 v + 1 = 0$$
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
$$a = 8$$
$$b = -2$$
$$c = 1$$
, then
D = b^2 - 4 * a * c =
(-2)^2 - 4 * (8) * (1) = -28
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
$$v_{1} = \frac{1}{8} + \frac{\sqrt{7} i}{8}$$
$$v_{2} = \frac{1}{8} - \frac{\sqrt{7} i}{8}$$
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:
$$x_{1} = $$
$$\frac{0}{1} + \frac{\left(\frac{1}{8} + \frac{\sqrt{7} i}{8}\right)^{\frac{1}{2}}}{1} = \sqrt{\frac{1}{8} + \frac{\sqrt{7} i}{8}}$$
$$x_{2} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(\frac{1}{8} + \frac{\sqrt{7} i}{8}\right)^{\frac{1}{2}}}{1} = - \sqrt{\frac{1}{8} + \frac{\sqrt{7} i}{8}}$$
$$x_{3} = $$
$$\frac{0}{1} + \frac{\left(\frac{1}{8} - \frac{\sqrt{7} i}{8}\right)^{\frac{1}{2}}}{1} = \sqrt{\frac{1}{8} - \frac{\sqrt{7} i}{8}}$$
$$x_{4} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(\frac{1}{8} - \frac{\sqrt{7} i}{8}\right)^{\frac{1}{2}}}{1} = - \sqrt{\frac{1}{8} - \frac{\sqrt{7} i}{8}}$$
Rapid solution
[src]
/ / ___\\ / / ___\\
4 ___ |atan\\/ 7 /| 4 ___ |atan\\/ 7 /|
\/ 2 *cos|-----------| I*\/ 2 *sin|-----------|
\ 2 / \ 2 /
x1 = - ---------------------- - ------------------------
2 2
$$x_{1} = - \frac{\sqrt[4]{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}$$
/ / ___\\ / / ___\\
4 ___ |atan\\/ 7 /| 4 ___ |atan\\/ 7 /|
\/ 2 *cos|-----------| I*\/ 2 *sin|-----------|
\ 2 / \ 2 /
x2 = - ---------------------- + ------------------------
2 2
$$x_{2} = - \frac{\sqrt[4]{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}$$
/ / ___\\ / / ___\\
4 ___ |atan\\/ 7 /| 4 ___ |atan\\/ 7 /|
\/ 2 *cos|-----------| I*\/ 2 *sin|-----------|
\ 2 / \ 2 /
x3 = ---------------------- - ------------------------
2 2
$$x_{3} = \frac{\sqrt[4]{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}$$
/ / ___\\ / / ___\\
4 ___ |atan\\/ 7 /| 4 ___ |atan\\/ 7 /|
\/ 2 *cos|-----------| I*\/ 2 *sin|-----------|
\ 2 / \ 2 /
x4 = ---------------------- + ------------------------
2 2
$$x_{4} = \frac{\sqrt[4]{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}$$
x4 = 2^(1/4)*cos(atan(sqrt(7))/2)/2 + 2^(1/4)*i*sin(atan(sqrt(7))/2)/2
Sum and product of roots
[src]
sum
/ / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\
4 ___ |atan\\/ 7 /| 4 ___ |atan\\/ 7 /| 4 ___ |atan\\/ 7 /| 4 ___ |atan\\/ 7 /| 4 ___ |atan\\/ 7 /| 4 ___ |atan\\/ 7 /| 4 ___ |atan\\/ 7 /| 4 ___ |atan\\/ 7 /|
\/ 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 /
- ---------------------- - ------------------------ + - ---------------------- + ------------------------ + ---------------------- - ------------------------ + ---------------------- + ------------------------
2 2 2 2 2 2 2 2
$$\left(\left(\frac{\sqrt[4]{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}\right) + \left(\left(- \frac{\sqrt[4]{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}\right) + \left(- \frac{\sqrt[4]{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}\right)\right)\right) + \left(\frac{\sqrt[4]{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}\right)$$
=
0
$$0$$
product
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ | 4 ___ |atan\\/ 7 /| 4 ___ |atan\\/ 7 /|| | 4 ___ |atan\\/ 7 /| 4 ___ |atan\\/ 7 /|| |4 ___ |atan\\/ 7 /| 4 ___ |atan\\/ 7 /|| |4 ___ |atan\\/ 7 /| 4 ___ |atan\\/ 7 /|| | \/ 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 /| |- ---------------------- - ------------------------|*|- ---------------------- + ------------------------|*|---------------------- - ------------------------|*|---------------------- + ------------------------| \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$\left(- \frac{\sqrt[4]{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}\right) \left(- \frac{\sqrt[4]{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}\right) \left(\frac{\sqrt[4]{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}\right) \left(\frac{\sqrt[4]{2} \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{2} i \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{7} \right)}}{2} \right)}}{2}\right)$$
=
1/8
$$\frac{1}{8}$$
1/8
Numerical answer
[src]
x1 = 0.489159171739258 + 0.338048362363489*i
x2 = 0.489159171739258 - 0.338048362363489*i
x3 = -0.489159171739258 + 0.338048362363489*i
x4 = -0.489159171739258 - 0.338048362363489*i
x4 = -0.489159171739258 - 0.338048362363489*i