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2 years ago

2n^3-3n^2+2n=oanswer needed urgently pls

Mathematics

1 answer:

Kazeer [188]2 years ago

3 0

$n = 0$

Step-by-step explanation:

<h2>Factor the expression</h2>

$2n ^ 3 - 3n ^ 2 + 2n = 0$

<h2>Separate into possible cases</h2>

$n(2n ^ 2 - 3n + 2) = 0$

<h2>Solve the equation</h2>

$n = 0[tex]n = 0 \: \: \: \: \: \: \: \: [tex]n = 0 \: \: \: \: \: \: \: \: 2n ^ 2 - 3n + 2 = 0$

<h2>Find the union</h2>

$n = 0[tex]n = 0 \: \: \: \: \: \: [tex]n = 0 \: \: \: \: \: \: n = R$

<h2>there for </h2>

$n = 0$

hope it help

#carry on learning

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4733

Step-by-step explanation:

Please refer to the attached diagram.

Point A can be assigned x-coordinate "p". Then its y-coordinate is 6p^2. The slope at that point is y'(p) = 12p.

Point B can be assigned x-coordinate "r". Then its y-coordinate is 6r^2. The slope at that point is y'(r) = 12r.

We want the slopes at those points to have a product of -1 (so the tangents are perpendicular). This means ...

(12p)(12r) = -1

r = -1/(144p)

The slope of line AB in the diagram is the ratio of the differences of y- and x-coordinates:

slope AB = (ry -py)/(rx -px) = (6r^2 -6p^2)/(r -p) = 6(r+p) . . . . simplified

The slope of AB is also the tangent of the sum of these angles: the angle AC makes with the x-axis and angle CAB. The tangent of a sum of angles is given by ...

tan(α+β) = (tan(α) +tan(β))/1 -tan(α)·tan(β))

Of course the slope of a line is equal to the tangent of the angle it makes with the x-axis. The tangent of angle CAB is 2 (because the aspect ratio of the rectangle is 2). This means we can write ...

slope AB = ((slope AC) +2)/(1 -(slope AC)(2))

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So, now we can figure the coordinates of points A and B, and the distance between them. That distance is given by the Pythagorean theorem as ...

d^2 = (6r^2 -6p^2)^2 +(r -p)^2

d^2 = (6(1/6)^2 -6(-1/24)^2)^2 +(1/6 +1/24)^2 = 25/1024 +25/576 = 625/9216

Because of the aspect ratio of the rectangle, the area is 2/5 of this value, so we have ...

Rectangle Area = (2/5)(625/9216) = 125/4608 = a/b

Then a+b = 125 +4608 = 4733.

_____

Comment on the solution

The point of intersection of the tangent lines is a fairly messy expression, and that propagates through any distance formulas used to find rectangle side lengths. This seemed much cleaner, though maybe not so obvious at first.