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 (A) 6x+4x-6=24+9x     simplify

10x - 6  = 24 + 9x       add 6 to both sides, subtract 9x from each side

x = 30    is trhe only solution     neither 

(B) 25-4x= 15-3x+10-x      simplify

25 - 4x   = 25 - 4x        identity  ....this is true for all x ...it has infinite solutions

 (C) 4x+8= 2x+7+2x-20       simplify

4x + 8 =  4x - 13         contradiction.......we can't add unlike quantites to equals and get equals.....hence, no solution

The ratio of two sisters ages is 2:1 = 36, therefore:

The younger sister is: 1/3 x 36 =12 years, and

The older sister is: 2/3 x 36 =24 years.

To use the Quadratic Formula.....From your PREVIOUS question, the quadratic equation MUST be in the form:

ax^2 + bx + c = 0

Your previous question was    8x^2 +3x - 11 = 691         which is NOT the correct form it has to equal 0

get it into proper form by subtracting 691 from each side

8x^2 + 3x - 702 =0

NOW  a = 8  b = 3  and c = -702   (NEGATIVE 702)

   a = 8   b = 3  and c = -702

x =  (-3 +- sqrt(9 - 4(8)(-702) )   )    /   2(8)

Need more help??

If so, please respond to THIS question....do not post another Forum entry.

Post the equations......we'll do our best.....!!!!

Let 'x' = sister's age...

then Kim's age = 2x

added together they equal 36     so:   x + 2x = 36     solve for 'x'

3x=36   (divide both sides by 3)

x=sister's age = 12    Kim's age = 2x = 24 

f(x)=(x+4)^2 + 3

The vertex of this function  is (-4, 3).........it will be one-to-one if we resrict the domain to either   (-inf, -4)  or (-4, inf).....x = -4 could belong to either original domain  and to either domain of the inverse

To find the inverse    write "y" for f(x)

y = (x + 4)^2 + 3      subtract 3 from both sides

y - 3   = ( x + 4)^2      take the pos/neg square roots of both sides

±√ [y - 3]  = x + 4      subtract 4 from both sides

±√ [y - 3] - 4  = x      "swap" x and y

±√ [x - 3] - 4  =  y        for y, write f^-1(x)

±√ [x - 3] - 4   =  f^-1 (x)

When we restrict the original domain to (-inf, -4),  -√ [x - 3] - 4 is the inverse

When we restrict the original domain to (-4, inf ) ,   √ [x - 3] - 4 is the inverse

Substituting, we have

x =   [ -3 ± √ [ (-3)^2   - 4 (8)(11]   /   [ 2 (8) ]

[ - 3 ± √  [ 9  - 352] / 16  =

[ -3 ± √ -343 ]  / 16  =

[ -3 ± √(  (-1) *49 * 7) ]  / 16 =

[ -3 ± 7 i √ 7 ] / 16 

Using Quadratic Formula    x = 9.1818     or   - 9.556

Quadratic Formula:

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

Thanks Alan,

I shall take a very good look at your answer :)

Congratulation GingerAle!!. You appear to have stumbled upon a very simple formula for generating the 2nd series!. I don't quite understand your modesty when you say "After hours of playing with it, I’ve not been able to make it work, either because my skills are poor –or because it really doesn’t work."

The 35th term of the 1st. one is: 35^3 - 35 =42,840

The 35th term of the 2nd. one is: 42,840/6=7,140. By using the formula provided by "Guest #4", "from one of Hade’s rivers: G_n(a_n)(z) = (z (-z^3+4 z^2-6 z+4))/(z-1)^4", I derive the 35th term of the this series as: 7,140!!!!!. I tried a couples more terms for both series and your suggestion for the 2nd series continues to hold true! Success!!.

Sorry, but the way this qustion is written, makes it very unclear as to how to solve it.

   ' 88% of books are classified by nonfiction or fiction.'    ..Are you sure you transcribed this part correctly?  Is there more info you have not given?

25 - 28  =  -3

1/5  is 'One FIfth'  =  1 divided by 5  =  .2

Pythagorean Theorem ONLY applies to RIGHT triangles. 

In an EQUALATERAL triangle you will have to draw a perpindicular  line from the MIDPOINT (that is a HINT) of the base to the apex to break it up in to TWO right triangles to calculate side lengths....

The calculator on this website will do this for you....look ABOVE the normal display and enter 144 / 15

48 / 5     9.6     or   9  3/5

The GCF of 30 and 42  is   6

Math only understand kid

Amazing! Or Amazing BS!

Guest #4 recognized the formula (g(n) = n³ - n) as a solution for generating these numbers (0, 6, 24, 60, 120, 210). (Heureka’s expansion makes this very clear.) 

However, the guest totally missed that this same formula, when divided by 6 ((g(n) =n³- n)/6), generates the solution for the second set of numbers) (0, 1, 4, 10, 20, 35) –maybe because the asker left out the (1) in the series.  He did pluck this convoluted mess from one of Hade’s rivers: G_n(a_n)(z) = (z (-z^3+4 z^2-6 z+4))/(z-1)^4. 

After hours of playing with it, I’ve not been able to make it work, either because my skills are poor –or because it really doesn’t work. 

Can anyone explain why this formula will or will not work for either of these series (0, 1, 4, 10, 20, 35) or (0, 4, 10, 20, 35)

 This is true

The result is zero:

\(\tan(x+45°)\rightarrow\frac{1+\tan x}{1-\tan x}\\\cot(x-45°)\rightarrow \frac{1+\tan x}{-1+\tan x}=-\tan(x+45°)\)

More generally:

.