Mister Exam

# x⁴-5x²+4=0 equation

A equation with variable:

#### Numerical solution:

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### The solution

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 4      2
x  - 5*x  + 4 = 0
$$\left(x^{4} - 5 x^{2}\right) + 4 = 0$$
Detail solution
Given the equation:
$$\left(x^{4} - 5 x^{2}\right) + 4 = 0$$
Do replacement
$$v = x^{2}$$
then the equation will be the:
$$v^{2} - 5 v + 4 = 0$$
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
$$a = 1$$
$$b = -5$$
$$c = 4$$
, then
D = b^2 - 4 * a * c =

(-5)^2 - 4 * (1) * (4) = 9

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