Mister Exam

# x⁴-3x²-4=0 equation

A equation with variable:

#### Numerical solution:

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### The solution

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 4      2
x  - 3*x  - 4 = 0
$$\left(x^{4} - 3 x^{2}\right) - 4 = 0$$
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*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) = 25

Because D > 0, then the equation has two roots.
v1 = (-b + sqrt(D)) / (2*a)

v2 = (-b - sqrt(D)) / (2*a)

or
$$v_{1} = 4$$
$$v_{2} = -1$$
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:
$$x_{1} =$$
$$\frac{0}{1} + \frac{4^{\frac{1}{2}}}{1} = 2$$
$$x_{2} =$$
$$\frac{\left(-1\right) 4^{\frac{1}{2}}}{1} + \frac{0}{1} = -2$$
$$x_{3} =$$
$$\frac{0}{1} + \frac{\left(-1\right)^{\frac{1}{2}}}{1} = i$$
$$x_{4} =$$
$$\frac{0}{1} + \frac{\left(-1\right) \left(-1\right)^{\frac{1}{2}}}{1} = - i$$
The graph
Sum and product of roots [src]
sum
-2 + 2 - I + I
$$\left(\left(-2 + 2\right) - i\right) + i$$
=
0
$$0$$
product
-2*2*(-I)*I
$$i - 4 \left(- i\right)$$
=
-4
$$-4$$
-4
Rapid solution [src]
x1 = -2
$$x_{1} = -2$$
x2 = 2
$$x_{2} = 2$$
x3 = -I
$$x_{3} = - i$$
x4 = I
$$x_{4} = i$$
x4 = i
x1 = -2.0
x2 = 2.0
x3 = 1.0*i
x4 = -1.0*i
x4 = -1.0*i