Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation sin(x)=4
-
Equation 5^x=125
-
Equation (|x+4|)=5
-
Equation 2*sin(x)=1
- Express {x} in terms of y where:
- 16*x-17*y=-15
- -4*x-4*y=19
- 15*x+1*y=19
- -5*x-2*y=9
-
Identical expressions
- x^ three - three .24x= zero
- x cubed minus 3.24x equally 0
- x to the power of three minus three .24x equally zero
- x3-3.24x=0
- x³-3.24x=0
- x to the power of 3-3.24x=0
- x^3-3.24x=O
-
Similar expressions
- x^3+3.24x=0
x^3-3.24x=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$x^{3} - \frac{81 x}{25} = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(x^{2} - \frac{81}{25}\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$x^{2} - \frac{81}{25} = 0$$
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
$$a = 1$$
$$b = 0$$
$$c = - \frac{81}{25}$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{2} = \frac{9}{5}$$
Simplify
$$x_{3} = - \frac{9}{5}$$
Simplify
The final answer for (x^3 - 81*x/25) + 0 = 0:
$$x_{1} = 0$$
$$x_{2} = \frac{9}{5}$$
$$x_{3} = - \frac{9}{5}$$
$$x^{3} - \frac{81 x}{25} = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(x^{2} - \frac{81}{25}\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$x^{2} - \frac{81}{25} = 0$$
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
$$a = 1$$
$$b = 0$$
$$c = - \frac{81}{25}$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (1) * (-81/25) = 324/25
Because D > 0, then the equation has two roots.
x2 = (-b + sqrt(D)) / (2*a)
x3 = (-b - sqrt(D)) / (2*a)
or
$$x_{2} = \frac{9}{5}$$
Simplify
$$x_{3} = - \frac{9}{5}$$
Simplify
The final answer for (x^3 - 81*x/25) + 0 = 0:
$$x_{1} = 0$$
$$x_{2} = \frac{9}{5}$$
$$x_{3} = - \frac{9}{5}$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - \frac{81}{25}$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{81}{25}$$
$$x_{1} x_{2} x_{3} = 0$$
$$p x^{2} + x^{3} + q x + v = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - \frac{81}{25}$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{81}{25}$$
$$x_{1} x_{2} x_{3} = 0$$
Rapid solution
[src]
x1 = -9/5
$$x_{1} = - \frac{9}{5}$$
x2 = 0
$$x_{2} = 0$$
x3 = 9/5
$$x_{3} = \frac{9}{5}$$
Sum and product of roots
[src]
sum
0 - 9/5 + 0 + 9/5
$$\left(\left(- \frac{9}{5} + 0\right) + 0\right) + \frac{9}{5}$$
=
0
$$0$$
product
1*-9/5*0*9/5
$$1 \left(- \frac{9}{5}\right) 0 \cdot \frac{9}{5}$$
=
0
$$0$$
0
The graph