Mister Exam

# x⁴-3x²-4=0 equation

The teacher will be very surprised to see your correct solution 😉

v

#### Numerical solution:

Do search numerical solution at [, ]

### The solution

You have entered [src]
 4      2
x  - 3*x  - 4 = 0
$$x^{4} - 3 x^{2} - 4 = 0$$
Detail solution
Given the equation:
$$x^{4} - 3 x^{2} - 4 = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 1$$
$$b = -3$$
$$c = -4$$
, then
$$D = b^2 - 4\ a\ c =$$
$$\left(-3\right)^{2} - 1 \cdot 4 \left(-4\right) = 25$$
Because D > 0, then the equation has two roots.
$$v_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$v_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$v_{1} = 4$$
Simplify
$$v_{2} = -1$$
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:
$$x_{1} = \frac{0}{1} + \frac{1 \cdot 4^{\frac{1}{2}}}{1} = 2$$
$$x_{2} = \frac{\left(-1\right) 4^{\frac{1}{2}}}{1} + \frac{0}{1} = -2$$
$$x_{3} = \frac{0}{1} + \frac{1 \left(-1\right)^{\frac{1}{2}}}{1} = i$$
$$x_{4} = \frac{0}{1} + \frac{\left(-1\right) \left(-1\right)^{\frac{1}{2}}}{1} = - i$$
The graph
Sum and product of roots [src]
sum
-2 + 2 + -I + I
$$\left(-2\right) + \left(2\right) + \left(- i\right) + \left(i\right)$$
=
0
$$0$$
product
-2 * 2 * -I * I
$$\left(-2\right) * \left(2\right) * \left(- i\right) * \left(i\right)$$
=
-4
$$-4$$
Rapid solution [src]
x_1 = -2
$$x_{1} = -2$$
x_2 = 2
$$x_{2} = 2$$
x_3 = -I
$$x_{3} = - i$$
x_4 = I
$$x_{4} = i$$
x1 = -1.0*i
x2 = 2.0
x3 = 1.0*i
x4 = -2.0
x4 = -2.0