Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation (x^2-1)/(x+2)=4x
-
Equation 3*x+5+(x+5)=(1-x)+4
-
Equation cos(x)=3/4
-
Equation cos(x)=(1/2)
- Express {x} in terms of y where:
- -13*x+7*y=11
- -8*x-15*y=15
- 4*x+18*y=-14
- -15*x-17*y=13
- Graphing y =:
- x^4-8x^2-16
-
Identical expressions
- x^ four -8x^ two - sixteen = zero
- x to the power of 4 minus 8x squared minus 16 equally 0
- x to the power of four minus 8x to the power of two minus sixteen equally zero
- x4-8x2-16=0
- x⁴-8x²-16=0
- x to the power of 4-8x to the power of 2-16=0
- x^4-8x^2-16=O
-
Similar expressions
- x^4+8x^2-16=0
- x^4-8x^2+16=0
x^4-8x^2-16=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(x^{4} - 8 x^{2}\right) - 16 = 0$$
Do replacement
$$v = x^{2}$$
then the equation will be the:
$$v^{2} - 8 v - 16 = 0$$
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
$$a = 1$$
$$b = -8$$
$$c = -16$$
, then
Because D > 0, then the equation has two roots.
or
$$v_{1} = 4 + 4 \sqrt{2}$$
$$v_{2} = 4 - 4 \sqrt{2}$$
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:
$$x_{1} = $$
$$\frac{0}{1} + \frac{\left(4 + 4 \sqrt{2}\right)^{\frac{1}{2}}}{1} = \sqrt{4 + 4 \sqrt{2}}$$
$$x_{2} = $$
$$\frac{\left(-1\right) \left(4 + 4 \sqrt{2}\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = - \sqrt{4 + 4 \sqrt{2}}$$
$$x_{3} = $$
$$\frac{0}{1} + \frac{\left(4 - 4 \sqrt{2}\right)^{\frac{1}{2}}}{1} = \sqrt{4 - 4 \sqrt{2}}$$
$$x_{4} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(4 - 4 \sqrt{2}\right)^{\frac{1}{2}}}{1} = - \sqrt{4 - 4 \sqrt{2}}$$
$$\left(x^{4} - 8 x^{2}\right) - 16 = 0$$
Do replacement
$$v = x^{2}$$
then the equation will be the:
$$v^{2} - 8 v - 16 = 0$$
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
$$a = 1$$
$$b = -8$$
$$c = -16$$
, then
D = b^2 - 4 * a * c =
(-8)^2 - 4 * (1) * (-16) = 128
Because D > 0, then the equation has two roots.
v1 = (-b + sqrt(D)) / (2*a)
v2 = (-b - sqrt(D)) / (2*a)
or
$$v_{1} = 4 + 4 \sqrt{2}$$
$$v_{2} = 4 - 4 \sqrt{2}$$
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:
$$x_{1} = $$
$$\frac{0}{1} + \frac{\left(4 + 4 \sqrt{2}\right)^{\frac{1}{2}}}{1} = \sqrt{4 + 4 \sqrt{2}}$$
$$x_{2} = $$
$$\frac{\left(-1\right) \left(4 + 4 \sqrt{2}\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = - \sqrt{4 + 4 \sqrt{2}}$$
$$x_{3} = $$
$$\frac{0}{1} + \frac{\left(4 - 4 \sqrt{2}\right)^{\frac{1}{2}}}{1} = \sqrt{4 - 4 \sqrt{2}}$$
$$x_{4} = $$
$$\frac{0}{1} + \frac{\left(-1\right) \left(4 - 4 \sqrt{2}\right)^{\frac{1}{2}}}{1} = - \sqrt{4 - 4 \sqrt{2}}$$
Sum and product of roots
[src]
sum
___________ ___________ ____________ ____________
/ ___ / ___ / ___ / ___
- 2*\/ 1 + \/ 2 + 2*\/ 1 + \/ 2 - 2*I*\/ -1 + \/ 2 + 2*I*\/ -1 + \/ 2
$$\left(\left(- 2 \sqrt{1 + \sqrt{2}} + 2 \sqrt{1 + \sqrt{2}}\right) - 2 i \sqrt{-1 + \sqrt{2}}\right) + 2 i \sqrt{-1 + \sqrt{2}}$$
=
0
$$0$$
product
___________ ___________ ____________ ____________
/ ___ / ___ / ___ / ___
-2*\/ 1 + \/ 2 *2*\/ 1 + \/ 2 *-2*I*\/ -1 + \/ 2 *2*I*\/ -1 + \/ 2
$$2 i \sqrt{-1 + \sqrt{2}} \cdot - 2 i \sqrt{-1 + \sqrt{2}} \cdot - 2 \sqrt{1 + \sqrt{2}} \cdot 2 \sqrt{1 + \sqrt{2}}$$
=
-16
$$-16$$
-16
Rapid solution
[src]
___________
/ ___
x1 = -2*\/ 1 + \/ 2
$$x_{1} = - 2 \sqrt{1 + \sqrt{2}}$$
___________
/ ___
x2 = 2*\/ 1 + \/ 2
$$x_{2} = 2 \sqrt{1 + \sqrt{2}}$$
____________
/ ___
x3 = -2*I*\/ -1 + \/ 2
$$x_{3} = - 2 i \sqrt{-1 + \sqrt{2}}$$
____________
/ ___
x4 = 2*I*\/ -1 + \/ 2
$$x_{4} = 2 i \sqrt{-1 + \sqrt{2}}$$
x4 = 2*i*sqrt(-1 + sqrt(2))
Numerical answer
[src]
x1 = 1.28718850581117*i
x2 = -1.28718850581117*i
x3 = 3.10754794806007
x4 = -3.10754794806007
x4 = -3.10754794806007