Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^4-x^3+5*x^2=0
-
Equation cos(x)=sqrt(2)/2
-
Equation x^2-6x+8=0
-
Equation 8(5-3x)=6(2-4x)+7
- Express {x} in terms of y where:
- 17*x-19*y=1
- -1*x+19*y=-10
- 10*x+9*y=-4
- 9*x-18*y=6
- Graphing y =:
- x^4-x^3+5*x^2
-
Identical expressions
- x^ four -x^ three + five *x^ two = zero
- x to the power of 4 minus x cubed plus 5 multiply by x squared equally 0
- x to the power of four minus x to the power of three plus five multiply by x to the power of two equally zero
- x4-x3+5*x2=0
- x⁴-x³+5*x²=0
- x to the power of 4-x to the power of 3+5*x to the power of 2=0
- x^4-x^3+5x^2=0
- x4-x3+5x2=0
- x^4-x^3+5*x^2=O
-
Similar expressions
- x^4-x^3-5*x^2=0
- x^4+x^3+5*x^2=0
x^4-x^3+5*x^2=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$5 x^{2} + \left(x^{4} - x^{3}\right) = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(x^{2} - x + 5\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$x^{2} - x + 5 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -1$$
$$c = 5$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{2} = \frac{1}{2} + \frac{\sqrt{19} i}{2}$$
$$x_{3} = \frac{1}{2} - \frac{\sqrt{19} i}{2}$$
The final answer for x^4 - x^3 + 5*x^2 = 0:
$$x_{1} = 0$$
$$x_{2} = \frac{1}{2} + \frac{\sqrt{19} i}{2}$$
$$x_{3} = \frac{1}{2} - \frac{\sqrt{19} i}{2}$$
$$5 x^{2} + \left(x^{4} - x^{3}\right) = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(x^{2} - x + 5\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$x^{2} - x + 5 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -1$$
$$c = 5$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (5) = -19
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x2 = (-b + sqrt(D)) / (2*a)
x3 = (-b - sqrt(D)) / (2*a)
or
$$x_{2} = \frac{1}{2} + \frac{\sqrt{19} i}{2}$$
$$x_{3} = \frac{1}{2} - \frac{\sqrt{19} i}{2}$$
The final answer for x^4 - x^3 + 5*x^2 = 0:
$$x_{1} = 0$$
$$x_{2} = \frac{1}{2} + \frac{\sqrt{19} i}{2}$$
$$x_{3} = \frac{1}{2} - \frac{\sqrt{19} i}{2}$$
Sum and product of roots
[src]
sum
____ ____ 1 I*\/ 19 1 I*\/ 19 - - -------- + - + -------- 2 2 2 2
$$\left(\frac{1}{2} - \frac{\sqrt{19} i}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{19} i}{2}\right)$$
=
1
$$1$$
product
/ ____\ / ____\ |1 I*\/ 19 | |1 I*\/ 19 | 0*|- - --------|*|- + --------| \2 2 / \2 2 /
$$0 \left(\frac{1}{2} - \frac{\sqrt{19} i}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{19} i}{2}\right)$$
=
0
$$0$$
0
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
____
1 I*\/ 19
x2 = - - --------
2 2
$$x_{2} = \frac{1}{2} - \frac{\sqrt{19} i}{2}$$
____
1 I*\/ 19
x3 = - + --------
2 2
$$x_{3} = \frac{1}{2} + \frac{\sqrt{19} i}{2}$$
x3 = 1/2 + sqrt(19)*i/2
Numerical answer
[src]
x1 = 0.5 - 2.17944947177034*i
x2 = 0.5 + 2.17944947177034*i
x3 = 0.0
x3 = 0.0
The graph