Mister Exam
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Similar expressions
- 2*x^4-16*x^2-32=0
- 2*x^4+16*x^2+32=0
2*x^4-16*x^2+32=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(2 x^{4} - 16 x^{2}\right) + 32 = 0$$
Do replacement
$$v = x^{2}$$
then the equation will be the:
$$2 v^{2} - 16 v + 32 = 0$$
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
$$a = 2$$
$$b = -16$$
$$c = 32$$
, then
Because D = 0, then the equation has one root.
$$v_{1} = 4$$
The final answer:
Because
$$v = x^{2}$$
then
$$x_{1} = \sqrt{v_{1}}$$
$$x_{2} = - \sqrt{v_{1}}$$
then:
$$x_{1} = $$
$$\frac{0}{1} + \frac{4^{\frac{1}{2}}}{1} = 2$$
$$x_{2} = $$
$$\frac{\left(-1\right) 4^{\frac{1}{2}}}{1} + \frac{0}{1} = -2$$
$$\left(2 x^{4} - 16 x^{2}\right) + 32 = 0$$
Do replacement
$$v = x^{2}$$
then the equation will be the:
$$2 v^{2} - 16 v + 32 = 0$$
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
$$a = 2$$
$$b = -16$$
$$c = 32$$
, then
D = b^2 - 4 * a * c =
(-16)^2 - 4 * (2) * (32) = 0
Because D = 0, then the equation has one root.
v = -b/2a = --16/2/(2)
$$v_{1} = 4$$
The final answer:
Because
$$v = x^{2}$$
then
$$x_{1} = \sqrt{v_{1}}$$
$$x_{2} = - \sqrt{v_{1}}$$
then:
$$x_{1} = $$
$$\frac{0}{1} + \frac{4^{\frac{1}{2}}}{1} = 2$$
$$x_{2} = $$
$$\frac{\left(-1\right) 4^{\frac{1}{2}}}{1} + \frac{0}{1} = -2$$