Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^2=12
-
Equation 12-x^2=11
-
Equation -sin(x)=0
-
Equation 4x+5=2x-7
- Express {x} in terms of y where:
- -5*x-6*y=-1
- -17*x-15*y=-17
- -2*x+7*y=-14
- -11*x-1*y=15
- Factor squared:
- x^4+2*x^2-1
-
Identical expressions
- x^ four + two *x^ two - one = zero
- x to the power of 4 plus 2 multiply by x squared minus 1 equally 0
- x to the power of four plus two multiply by x to the power of two minus one equally zero
- x4+2*x2-1=0
- x⁴+2*x²-1=0
- x to the power of 4+2*x to the power of 2-1=0
- x^4+2x^2-1=0
- x4+2x2-1=0
- x^4+2*x^2-1=O
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Similar expressions
- x^4+2*x^2+1=0
- x^4-2*x^2-1=0
x^4+2*x^2-1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(x^{4} + 2 x^{2}\right) - 1 = 0$$
Do replacement
$$v = x^{2}$$
then the equation will be the:
$$v^{2} + 2 v - 1 = 0$$
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
$$a = 1$$
$$b = 2$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$v_{1} = -1 + \sqrt{2}$$
$$v_{2} = - \sqrt{2} - 1$$
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:
$$x_{1} = $$
$$\frac{0}{1} + \frac{\left(-1 + \sqrt{2}\right)^{\frac{1}{2}}}{1} = \sqrt{-1 + \sqrt{2}}$$
$$x_{2} = $$
$$\frac{\left(-1\right) \left(-1 + \sqrt{2}\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = - \sqrt{-1 + \sqrt{2}}$$
$$x_{3} = $$
$$\frac{0}{1} + \frac{\left(- \sqrt{2} - 1\right)^{\frac{1}{2}}}{1} = \sqrt{- \sqrt{2} - 1}$$
$$x_{4} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(- \sqrt{2} - 1\right)^{\frac{1}{2}}}{1} = - \sqrt{- \sqrt{2} - 1}$$
$$\left(x^{4} + 2 x^{2}\right) - 1 = 0$$
Do replacement
$$v = x^{2}$$
then the equation will be the:
$$v^{2} + 2 v - 1 = 0$$
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
$$a = 1$$
$$b = 2$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(2)^2 - 4 * (1) * (-1) = 8
Because D > 0, then the equation has two roots.
v1 = (-b + sqrt(D)) / (2*a)
v2 = (-b - sqrt(D)) / (2*a)
or
$$v_{1} = -1 + \sqrt{2}$$
$$v_{2} = - \sqrt{2} - 1$$
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:
$$x_{1} = $$
$$\frac{0}{1} + \frac{\left(-1 + \sqrt{2}\right)^{\frac{1}{2}}}{1} = \sqrt{-1 + \sqrt{2}}$$
$$x_{2} = $$
$$\frac{\left(-1\right) \left(-1 + \sqrt{2}\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = - \sqrt{-1 + \sqrt{2}}$$
$$x_{3} = $$
$$\frac{0}{1} + \frac{\left(- \sqrt{2} - 1\right)^{\frac{1}{2}}}{1} = \sqrt{- \sqrt{2} - 1}$$
$$x_{4} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(- \sqrt{2} - 1\right)^{\frac{1}{2}}}{1} = - \sqrt{- \sqrt{2} - 1}$$
Rapid solution
[src]
____________
/ ___
x1 = -\/ -1 + \/ 2
$$x_{1} = - \sqrt{-1 + \sqrt{2}}$$
____________
/ ___
x2 = \/ -1 + \/ 2
$$x_{2} = \sqrt{-1 + \sqrt{2}}$$
___________
/ ___
x3 = -I*\/ 1 + \/ 2
$$x_{3} = - i \sqrt{1 + \sqrt{2}}$$
___________
/ ___
x4 = I*\/ 1 + \/ 2
$$x_{4} = i \sqrt{1 + \sqrt{2}}$$
x4 = i*sqrt(1 + sqrt(2))
Sum and product of roots
[src]
sum
____________ ____________ ___________ ___________
/ ___ / ___ / ___ / ___
- \/ -1 + \/ 2 + \/ -1 + \/ 2 - I*\/ 1 + \/ 2 + I*\/ 1 + \/ 2
$$\left(\left(- \sqrt{-1 + \sqrt{2}} + \sqrt{-1 + \sqrt{2}}\right) - i \sqrt{1 + \sqrt{2}}\right) + i \sqrt{1 + \sqrt{2}}$$
=
0
$$0$$
product
____________ ____________ / ___________\ ___________ / ___ / ___ | / ___ | / ___ -\/ -1 + \/ 2 *\/ -1 + \/ 2 *\-I*\/ 1 + \/ 2 /*I*\/ 1 + \/ 2
$$i \sqrt{1 + \sqrt{2}} \cdot - i \sqrt{1 + \sqrt{2}} \cdot - \sqrt{-1 + \sqrt{2}} \sqrt{-1 + \sqrt{2}}$$
=
-1
$$-1$$
-1
Numerical answer
[src]
x1 = 1.55377397403004*i
x2 = -0.643594252905583
x3 = 0.643594252905583
x4 = -1.55377397403004*i
x4 = -1.55377397403004*i