Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation cos(x)-1=0
-
Equation 3*x=0
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Equation x+5=7
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Equation 16*x^3+8*x^2+x=0
- Express {x} in terms of y where:
- 19*x+2*y=17
- 8*x-9*y=-1
- 13*x-1*y=-20
- 9*x+8*y=-4
- Factor polynomial:
- 16*x^3+8*x^2+x
-
Identical expressions
- sixteen *x^ three + eight *x^ two +x= zero
- 16 multiply by x cubed plus 8 multiply by x squared plus x equally 0
- sixteen multiply by x to the power of three plus eight multiply by x to the power of two plus x equally zero
- 16*x3+8*x2+x=0
- 16*x³+8*x²+x=0
- 16*x to the power of 3+8*x to the power of 2+x=0
- 16x^3+8x^2+x=0
- 16x3+8x2+x=0
- 16*x^3+8*x^2+x=O
-
Similar expressions
- 16*x^3+8*x^2-x=0
- 16*x^3-8*x^2+x=0
16*x^3+8*x^2+x=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$x + \left(16 x^{3} + 8 x^{2}\right) = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(16 x^{2} + 8 x + 1\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$16 x^{2} + 8 x + 1 = 0$$
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
$$a = 16$$
$$b = 8$$
$$c = 1$$
, then
Because D = 0, then the equation has one root.
$$x_{2} = - \frac{1}{4}$$
The final answer for 16*x^3 + 8*x^2 + x = 0:
$$x_{1} = 0$$
$$x_{2} = - \frac{1}{4}$$
$$x + \left(16 x^{3} + 8 x^{2}\right) = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(16 x^{2} + 8 x + 1\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$16 x^{2} + 8 x + 1 = 0$$
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
$$a = 16$$
$$b = 8$$
$$c = 1$$
, then
D = b^2 - 4 * a * c =
(8)^2 - 4 * (16) * (1) = 0
Because D = 0, then the equation has one root.
x = -b/2a = -8/2/(16)
$$x_{2} = - \frac{1}{4}$$
The final answer for 16*x^3 + 8*x^2 + x = 0:
$$x_{1} = 0$$
$$x_{2} = - \frac{1}{4}$$
Vieta's Theorem
rewrite the equation
$$x + \left(16 x^{3} + 8 x^{2}\right) = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{x^{2}}{2} + \frac{x}{16} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{1}{2}$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{16}$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \frac{1}{2}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{1}{16}$$
$$x_{1} x_{2} x_{3} = 0$$
$$x + \left(16 x^{3} + 8 x^{2}\right) = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + \frac{x^{2}}{2} + \frac{x}{16} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{1}{2}$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{16}$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - \frac{1}{2}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{1}{16}$$
$$x_{1} x_{2} x_{3} = 0$$
Sum and product of roots
[src]
sum
-1/4
$$- \frac{1}{4}$$
=
-1/4
$$- \frac{1}{4}$$
product
0*(-1) ------ 4
$$\frac{\left(-1\right) 0}{4}$$
=
0
$$0$$
0
The graph