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sertanlavr [38]
3 years ago
13

Six more than the quotient of a number and 2 equals 8

Mathematics
2 answers:
ikadub [295]3 years ago
7 0
Let number=x

(X/2)+6=8

First subtract 6 on each side to isolate X on one side.

(X/2)=2

Now to fully isolate X by itself, just multiply 2 on each side.

X=4
astraxan [27]3 years ago
6 0

Answer:

Step-by-step explanation:6 + x/2 = 8

6-6 +x/2 =8-6

x/2 = 2

x/2(2) = (2)2

x=4

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Question (c)! How do I know that t^5-10t^3+5t=0?<br> Thanks!
astra-53 [7]
(a) By DeMoivre's theorem, we have

(\cos\theta+i\sin\theta)^5=\cos5\theta+i\sin5\theta

On the LHS, expanding yields

\cos^5\theta+5i\cos^4\theta\sin\theta-10\cos^3\theta\sin^2\theta-10i\cos^2\theta\sin^3\theta+5\cos\theta\sin^4\theta+i\sin^4\theta

Matching up real and imaginary parts, we have for (i) and (ii),


\cos5\theta=\cos^5\theta-10\cos^3\theta\sin^2\theta+5\cos\theta\sin^4\theta
\sin5\theta=5\cos^4\theta\sin\theta-10\cos^2\theta\sin^3\theta+\sin^5\theta

(b) By the definition of the tangent function,

\tan5\theta=\dfrac{\sin5\theta}{\cos5\theta}
=\dfrac{5\cos^4\theta\sin\theta-10\cos^2\theta\sin^3\theta+\sin^5\theta}{\cos^5\theta-10\cos^3\theta\sin^2\theta+5\cos\theta\sin^4\theta}

=\dfrac{5\tan\theta-10\tan^3\theta+\tan^5\theta}{1-10\tan^2\theta+5\tan^4\theta}
=\dfrac{t^5-10t^3+5t}{5t^4-10t^2+1}


(c) Setting \theta=\dfrac\pi5, we have t=\tan\dfrac\pi5 and \tan5\left(\dfrac\pi5\right)=\tan\pi=0. So

0=\dfrac{t^5-10t^3+5t}{5t^4-10t^2+1}

At the given value of t, the denominator is a non-zero number, so only the numerator can contribute to this reducing to 0.


0=t^5-10t^3+5t\implies0=t^4-10t^2+5

Remember, this is saying that

0=\tan^4\dfrac\pi5-10\tan^2\dfrac\pi5+5

If we replace \tan^2\dfrac\pi5 with a variable x, then the above means \tan^2\dfrac\pi5 is a root to the quadratic equation,

x^2-10x+5=0

Also, if \theta=\dfrac{2\pi}5, then t=\tan\dfrac{2\pi}5 and \tan5\left(\dfrac{2\pi}5\right)=\tan2\pi=0. So by a similar argument as above, we deduce that \tan^2\dfrac{2\pi}5 is also a root to the quadratic equation above.

(d) We know both roots to the quadratic above. The fundamental theorem of algebra lets us write

x^2-10x+5=\left(x-\tan^2\dfrac\pi5\right)\left(x-\tan^2\dfrac{2\pi}5\right)

Expand the RHS and match up terms of the same power. In particular, the constant terms satisfy

5=\tan^2\dfrac\pi5\tan^2\dfrac{2\pi}5\implies\tan\dfrac\pi5\tan\dfrac{2\pi}5=\pm\sqrt5

But \tanx>0 for all 0, as is the case for x=\dfrac\pi5 and x=\dfrac{2\pi}5, so we choose the positive root.
3 0
3 years ago
COMPLETE<br> The equation X-9=0 has how many <br> real solution(s).
Maru [420]

Answer:

one solution

Step-by-step explanation:

X-9=0

Add 9 to each side

X-9+9=0+9

x = 9

There is one solution

5 0
3 years ago
The base of 40-foot ladder is 8 feet from the wall. How high is the ladder on the wall (round to the nearest foot)
Nina [5.8K]

Given:

The base of 40-foot ladder is 8 feet from the wall.

To find:

How high is the ladder on the wall (round to the nearest foot).

Solution:

Ladder makes a right angle triangle with wall and ground.

We have,

Length of ladder (hypotenuse)= 40 foot

Base = 8 foot

We need to find the perpendicular to get the height of the ladder on the wall.

Let h be the height of the ladder on the wall.

According to the Pythagoras theorem,

Hypotenuse^2=Base^2+Perpendicular^2

(40)^2=(8)^2+(h)^2

1600=64+h^2

1600-64=h^2

1536=h^2

Taking square root on both sides.

\pm \sqrt{1536}=h

\pm 39.1918358=h

Height cannot be negative. Round to the nearest foot.

h\approx 39

Therefore, the height of the ladder on the wall is 39 foot.

5 0
3 years ago
45% of the load is 27 tonnes. Find the full load.
VashaNatasha [74]

the full load is 60 tonnes

5 0
3 years ago
Help please<br><br> Factorise<br><br> x²-x-56<br><br> its due in 4 minutes
Iteru [2.4K]

Answer:

(x-8)(x+7)

Step-by-step explanation:

using quadratic formula we have

x=  \frac{-b+-\sqrt{b^{2}-4*a*c}}{2*a}

we have:

x^2-x-56=0

a= 1 b=-1 c=-56

so we have:

x=(-(-1)+-sqrt(1^1-4*(1)*(-56)))/(2*1)

x=1+-sqrt(1-4*(-56))/(2)

x=(1+-sqrt(225))/2

x=(1+-15)/2

so we have the roots:

x1=(1+15)/2 =8

x2=(1-15)/2=-7

so the answer is (x-8)(x+7)

7 0
3 years ago
Read 2 more answers
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