Mister Exam

# x^4+3x^2-10=0 equation

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#### Numerical solution:

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### The solution

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 4      2
x  + 3*x  - 10 = 0
$$x^{4} + 3 x^{2} - 10 = 0$$
Detail solution
Given the equation:
$$x^{4} + 3 x^{2} - 10 = 0$$
Do replacement
$$v = x^{2}$$
then the equation will be the:
$$v^{2} + 3 v - 10 = 0$$
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
$$a = 1$$
$$b = 3$$
$$c = -10$$
, then
D = b^2 - 4 * a * c =

(3)^2 - 4 * (1) * (-10) = 49

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

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

or
$$v_{1} = 2$$
Simplify
$$v_{2} = -5$$
Simplify
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:
$$x_{1} = \frac{0}{1} + \frac{1 \cdot 2^{\frac{1}{2}}}{1} = \sqrt{2}$$
$$x_{2} = \frac{\left(-1\right) 2^{\frac{1}{2}}}{1} + \frac{0}{1} = - \sqrt{2}$$
$$x_{3} = \frac{0}{1} + \frac{1 \left(-5\right)^{\frac{1}{2}}}{1} = \sqrt{5} i$$
$$x_{4} = \frac{0}{1} + \frac{\left(-1\right) \left(-5\right)^{\frac{1}{2}}}{1} = - \sqrt{5} i$$
The graph
Sum and product of roots [src]
sum
      ___     ___       ___       ___
0 - \/ 2  + \/ 2  - I*\/ 5  + I*\/ 5 
$$\left(\left(\left(- \sqrt{2} + 0\right) + \sqrt{2}\right) - \sqrt{5} i\right) + \sqrt{5} i$$
=
0
$$0$$
product
     ___   ___      ___     ___
1*-\/ 2 *\/ 2 *-I*\/ 5 *I*\/ 5 
$$\sqrt{5} i \sqrt{2} \cdot 1 \left(- \sqrt{2}\right) \left(- \sqrt{5} i\right)$$
=
-10
$$-10$$
-10
Rapid solution [src]
        ___
x1 = -\/ 2 
$$x_{1} = - \sqrt{2}$$
       ___
x2 = \/ 2 
$$x_{2} = \sqrt{2}$$
          ___
x3 = -I*\/ 5 
$$x_{3} = - \sqrt{5} i$$
         ___
x4 = I*\/ 5 
$$x_{4} = \sqrt{5} i$$
x1 = -1.4142135623731
x2 = 2.23606797749979*i
x3 = 1.4142135623731
x4 = -2.23606797749979*i
x4 = -2.23606797749979*i