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Sum = F / [1 - R], where F=First term, R = Common Ratio

Ratio = (1/9) / 2 =(1/9 x 1/2) =1 / 18

Sum = 2 / [1 - 1/18]

Sum = 2 / [17/18]

Sum = 2 x [18/17]

Sum = 36 / 17

5a^2 = 320      divide both sides by 5

a^2 = 64          take the square root of both sides

a =sqrt(64)

a = +8 or -8

That's a good way MIRB, I didn't know of that!

You can also do it by typing:  abs(-2)

Question answered here: https://web2.0calc.com/questions/math-postage-stamp-problem

original area   =   L * W

new length   =   L + 20% * L   =   L + 0.2L   =   1.2L

new width   =   W + 10% * W   =   W + 0.1W   =   1.1W

new area   =   1.2L * 1.1W

new area   =   1.2 * 1.1 * L * W

new area   =   1.32 * L * W

new area   =   1.32 * original area

new area   =   (1 + 0.32) * original area

new area   =   original area + 0.32 * original area

new area   =   original area + 32% * original area

The new area is the original area increased by  32%  of the original area.

Yes, press shift+backslash to make this: |

Surround a number with the absolute value signs, such as this: |-2|

The calculator should spit out 2

1.

The maximum  y  value is  1  and the minumum  y  value is  -7

The equation of the midline is   y  =  (1 + -7)/2   =   -6/2   =  -3

The equation of the midline is  y  =  -3

2.

The maximum  y  value is  6   and the minimum  y  value is  -4 .

The amplitude is   (6 - -4)/2   =   10/2   =   5        

No one can answer this unless you specify the tax brackets or any other relevant thing. 

Remember this is an international forum!

I most likely live on the other side of the world to you and the tax system is quite different.

40 hours x $15.20 = $608 - what Tim earned in regular pay.

51 hours - 40 hours =11 hours of overtime he worked.

Since you don't specify what the "usual rate for overtime" is, we can only check the gross pay.

Since $608 and $775.20 are @ $15.20 for 40 and 51 hours respectively, then will have to check the next amount of $858.80.

$858.80 - $608 =250.80 - this is overtime pay that Tim got. And we check to see the overtime rate:

$250.80 / 11 hours =$22.80 per hour for 11 hours of overtime, which is:

$22.80 / $15.20 =1.5 times regular pay, which is very common. Therefore Tim's gross pay is:

40 regular hours x $15.20 =$608 Tim's regular pay

11 hours of overtime x [1.5 x $15.20] =$250.80 Tim's ovetime pay.

$608.00 + 250.80 = $858.80 Tim's gross pay.

f(x)  =  (x - 3)/(x - 4)

                                     First let's find  f^-1(x) . Instead of  f(x) , write  y .

y   =   (x - 3)/(x - 4)

                                     Now solve for  x . Multiply both sides of the equation by  (x - 4) .

y(x - 4)   =   x - 3

xy - 4y   =   x - 3

                                    Add  4y  to both sides and subtract  x  from both sides.

xy - x   =   4y - 3

                                    Factor  x  out of the terms on the left side.

x(y - 1)   =   4y - 3

                                    Divide both sides by  (y - 1) .

x   =   (4y - 3)/(y - 1)

                                    So...

f^-1(x)  =  (4x - 3)/(x - 1)

And...

(4x - 3)/(x - 1)  is undefined when  x - 1  =  0

f^-1(x)  is undefined when  x = 1

Guest, this has a degree of 4 so there are 4 roots, some may be double or multiple roots of course.

The graph is concave up because the leading coefficent is +9 (positive)

\(9x^4-12x^3+28x^2-16x+16 \)

I do not think guest has explained about writing this as a square.

I suspect he used a calculator program to do that.

However I can see that 9x^4 and 16 are both perfect squares so it is not unreasonable to examine the squared possibility.

It would have to be of the form    \((3x^2+ax+4)^2\)   if it is a perfect square.

\((3x^2+ax+4)^2\\ =3x^2(3x^2+ax+4)+ax(3x^2+ax+4)+4(3x^2+ax+4)\\ =9x^4+3ax^3+12x^2+3ax^3+a^2x^2+4ax+12x^2+4ax+16\\ =9x^4+3ax^3+3ax^3+12x^2+a^2x^2+12x^2+4ax+4ax+16\\ =9x^4\qquad+6ax^3\qquad+(24+a^2)x^2\qquad\qquad+8ax\qquad+16\\\)

6a=-12 so a =-2

so

24+(-2)^2=28  good

8a=-16            good

so

\(9x^4-12x^3+28x^2-16x+16 \) = \((3x^2-2x+4)^2\)

So the roots would be the positive and negative of the answers that our guest found.

(Assuming s/he made no mistakes)

ie

\(x =\pm\left(\frac{ 1+ i \sqrt{11}}{3}\right)      \qquad   or  \qquad        x =\pm\left(\frac{ 1- i \sqrt{11}}{3} \right)\)

These triangles are not only congruent they are drawn to look congruent!

If you look at the pics you can discard 3 of the answers without even knowing any rules.

Then again, do you know what congruent means?

If not then google it!!

Solve for x:

9 x^4 - 12 x^3 + 28 x^2 - 16 x + 16 = 0

Write the left hand side as a square:

(3 x^2 - 2 x + 4)^2 = 0

Take the square root of both sides:

3 x^2 - 2 x + 4 = 0

Divide both sides by 3:

x^2 - (2 x)/3 + 4/3 = 0

Subtract 4/3 from both sides:

x^2 - (2 x)/3 = -4/3

Add 1/9 to both sides:

x^2 - (2 x)/3 + 1/9 = -11/9

Write the left hand side as a square:

(x - 1/3)^2 = -11/9

Take the square root of both sides:

x - 1/3 = (i sqrt(11))/3 or x - 1/3 = 1/3 (-i) sqrt(11)

Add 1/3 to both sides:

x = 1/3 + (i sqrt(11))/3 or x - 1/3 = 1/3 (-i) sqrt(11)

Add 1/3 to both sides:

x = 1/3 + (i sqrt(11))/3         or          x = 1/3 - (i sqrt(11))/3

There is STILL no solution the way you have the equations written down. However, if you modify the signs, + or - on the first and the fourth, then you have this solution:

Solve the following system:

{a = -a + 2 + sqrt(5) | (equation 1)

b = -b + 2 - sqrt(5) | (equation 2)

c = -c + 2 + sqrt(5) | (equation 3)

d = -d + 2 - sqrt(5) | (equation 4)

Express the system in standard form:

{2 a+0 b+0 c+0 d = 2 + sqrt(5) | (equation 1)

0 a+2 b+0 c+0 d = 2 - sqrt(5) | (equation 2)

0 a+0 b+2 c+0 d = 2 + sqrt(5) | (equation 3)

0 a+0 b+0 c+2 d = 2 - sqrt(5) | (equation 4)

Divide equation 4 by 2:

{2 a+0 b+0 c+0 d = 2 + sqrt(5) | (equation 1)

0 a+2 b+0 c+0 d = 2 - sqrt(5) | (equation 2)

0 a+0 b+2 c+0 d = 2 + sqrt(5) | (equation 3)

0 a+0 b+0 c+d = 1 - (sqrt(5))/(2) | (equation 4)

Divide equation 3 by 2:

{2 a+0 b+0 c+0 d = 2 + sqrt(5) | (equation 1)

0 a+2 b+0 c+0 d = 2 - sqrt(5) | (equation 2)

0 a+0 b+c+0 d = (sqrt(5) + 2)/(2) | (equation 3)

0 a+0 b+0 c+d = 1 - sqrt(5)/2 | (equation 4)

Divide equation 2 by 2:

{2 a+0 b+0 c+0 d = 2 + sqrt(5) | (equation 1)

0 a+b+0 c+0 d = 1 - (sqrt(5))/(2) | (equation 2)

0 a+0 b+c+0 d = 1/2 (2 + sqrt(5)) | (equation 3)

0 a+0 b+0 c+d = 1 - sqrt(5)/2 | (equation 4)

Divide equation 1 by 2:

{a+0 b+0 c+0 d = (sqrt(5) + 2)/(2) | (equation 1)

0 a+b+0 c+0 d = 1 - sqrt(5)/2 | (equation 2)

0 a+0 b+c+0 d = 1/2 (2 + sqrt(5)) | (equation 3)

0 a+0 b+0 c+d = 1 - sqrt(5)/2 | (equation 4)

Collect results:

a = 1/2 (2 + sqrt(5))

b = 1 - sqrt(5)/2

c = 1/2 (2 + sqrt(5))

d = 1 - sqrt(5)/2      If you multiply them together, you get:abcd =1/16

YZ  is formed by the angle marked by one red line.

JG  is formed by the angle marked by no red lines.

So we cannot say that YZ = JG .

XY is formed by the angle marked by two red lines.

JH  is formed by the angle marked by one red line.

So we cannot say that  XY = JH

∠Y is marked by no red lines.

∠J  is marked by two red lines.

So we cannot say that ∠Y = ∠J

∠Y is marked by no red lines.

∠H is marked by no red lines.

So ∠Y = ∠H

Sorry, I changed the question

Will do

Your arithmetic series as written is missing something, because n comes out as a non-integer. Check your numbers carefully.