Mister Exam
Other calculators
-
How to use it?
- Equation:
- Equation x-y=7
-
Equation 1/(x+6)=2
-
Equation (x-4)^2-x^2=0
-
Equation 6/(x+5)=-5
- Express {x} in terms of y where:
- 7*x-1*y=-7
- 10*x-15*y=16
- -14*x-12*y=5
- 13*x-12*y=19
- Graphing y =:
- x^3-6*x^2+16
-
Identical expressions
- x^ three - six *x^ two + sixteen = zero
- x cubed minus 6 multiply by x squared plus 16 equally 0
- x to the power of three minus six multiply by x to the power of two plus sixteen equally zero
- x3-6*x2+16=0
- x³-6*x²+16=0
- x to the power of 3-6*x to the power of 2+16=0
- x^3-6x^2+16=0
- x3-6x2+16=0
- x^3-6*x^2+16=O
-
Similar expressions
- x^3-6*x^2-16=0
- x^3+6*x^2+16=0
x^3-6*x^2+16=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(x^{3} - 6 x^{2}\right) + 16 = 0$$
transform
$$\left(- 6 x^{2} + \left(x^{3} - 8\right)\right) + 24 = 0$$
or
$$\left(- 6 x^{2} + \left(x^{3} - 2^{3}\right)\right) + 6 \cdot 2^{2} = 0$$
$$- 6 \left(x^{2} - 2^{2}\right) + \left(x^{3} - 2^{3}\right) = 0$$
$$- 6 \left(x - 2\right) \left(x + 2\right) + \left(x - 2\right) \left(\left(x^{2} + 2 x\right) + 2^{2}\right) = 0$$
Take common factor -2 + x from the equation
we get:
$$\left(x - 2\right) \left(- 6 \left(x + 2\right) + \left(\left(x^{2} + 2 x\right) + 2^{2}\right)\right) = 0$$
or
$$\left(x - 2\right) \left(x^{2} - 4 x - 8\right) = 0$$
then:
$$x_{1} = 2$$
and also
we get the equation
$$x^{2} - 4 x - 8 = 0$$
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
$$a = 1$$
$$b = -4$$
$$c = -8$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{2} = 2 + 2 \sqrt{3}$$
$$x_{3} = 2 - 2 \sqrt{3}$$
The final answer for x^3 - 6*x^2 + 16 = 0:
$$x_{1} = 2$$
$$x_{2} = 2 + 2 \sqrt{3}$$
$$x_{3} = 2 - 2 \sqrt{3}$$
$$\left(x^{3} - 6 x^{2}\right) + 16 = 0$$
transform
$$\left(- 6 x^{2} + \left(x^{3} - 8\right)\right) + 24 = 0$$
or
$$\left(- 6 x^{2} + \left(x^{3} - 2^{3}\right)\right) + 6 \cdot 2^{2} = 0$$
$$- 6 \left(x^{2} - 2^{2}\right) + \left(x^{3} - 2^{3}\right) = 0$$
$$- 6 \left(x - 2\right) \left(x + 2\right) + \left(x - 2\right) \left(\left(x^{2} + 2 x\right) + 2^{2}\right) = 0$$
Take common factor -2 + x from the equation
we get:
$$\left(x - 2\right) \left(- 6 \left(x + 2\right) + \left(\left(x^{2} + 2 x\right) + 2^{2}\right)\right) = 0$$
or
$$\left(x - 2\right) \left(x^{2} - 4 x - 8\right) = 0$$
then:
$$x_{1} = 2$$
and also
we get the equation
$$x^{2} - 4 x - 8 = 0$$
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
$$a = 1$$
$$b = -4$$
$$c = -8$$
, then
D = b^2 - 4 * a * c =
(-4)^2 - 4 * (1) * (-8) = 48
Because D > 0, then the equation has two roots.
x2 = (-b + sqrt(D)) / (2*a)
x3 = (-b - sqrt(D)) / (2*a)
or
$$x_{2} = 2 + 2 \sqrt{3}$$
$$x_{3} = 2 - 2 \sqrt{3}$$
The final answer for x^3 - 6*x^2 + 16 = 0:
$$x_{1} = 2$$
$$x_{2} = 2 + 2 \sqrt{3}$$
$$x_{3} = 2 - 2 \sqrt{3}$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -6$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = 16$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 6$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = 16$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -6$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = 16$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 6$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = 16$$
Rapid solution
[src]
x1 = 2
$$x_{1} = 2$$
___ x2 = 2 - 2*\/ 3
$$x_{2} = 2 - 2 \sqrt{3}$$
___ x3 = 2 + 2*\/ 3
$$x_{3} = 2 + 2 \sqrt{3}$$
x3 = 2 + 2*sqrt(3)
Sum and product of roots
[src]
sum
___ ___ 2 + 2 - 2*\/ 3 + 2 + 2*\/ 3
$$\left(\left(2 - 2 \sqrt{3}\right) + 2\right) + \left(2 + 2 \sqrt{3}\right)$$
=
6
$$6$$
product
/ ___\ / ___\ 2*\2 - 2*\/ 3 /*\2 + 2*\/ 3 /
$$2 \left(2 - 2 \sqrt{3}\right) \left(2 + 2 \sqrt{3}\right)$$
=
-16
$$-16$$
-16
The graph