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2(-1)^2 -8 (-1) + 1   =  11

2(2)^2 - 8(2)+ 1 =  -7

Range =  [ -7, 11 ]

\(f(x) = \frac{1}{1+\frac{2}{1+\frac 3x}}. \)

x cannot  = 0

1 + 3/x  =  [ x + 3] / x

2 /  [ ( x + 3) /x ] =   2x / [x + 3]

x cannot = -3

1 + 2x  / [ x + 3]  =   [ x + 3 + 2x ] / [ x + 3] = [ 3x + 3] / [x + 3]

1/ ([3x + 3] /[x + 3]) =    [x + 3] / [ 3x + 3]

x cannot = -1

(c) Does there exist a combination of red potion and blue potion that can produce a potion that is 75% magical syrup by volume?

Not possible since neither potion exceeds 60 %  magical syrup

(b) Find the amounts of red potion and blue potion (in mL) that can be combined in order to produce 100 mL of a potion that is 54 magical syrup by volume.

Let x be the amount of red potion and 100-x be the amount of blue

.60x + .30 (100-x)  = .54 * 100

.60x + 30  - .30x  = 54

.30x + 30 = 54

.30x = 24

x = 24/.30  = 80

80ml of red must be added

20 ml of blue must be added

(a) Find the amount of red potion (in mL) that must be added to 500 mL of blue potion in order to produce potion that is 40% magical syrup by volume.

Let x be the amount of red potion to  be added

.60x  + 500 * .30  =   (500 + x) *.40

.60x + 150  = .200 + .40x

,60x -.40x = 200 - 150

.20x = 50

x = 50 /.20  = 250ml  of red must be added

https://web2.0calc.com/questions/algebra_55993

Simplify as

x^2 + 12x - 4   = 0

Let the roots = a,b

Sum of the  roots  =  a + b =  -12

Product of the roots = -4  = ab  →   -8 = 2ab

a + b  = -12        square both sides

a^2 + 2ab + b^2 = 144

a^2  - 8 + b^2 =144

a^2 + b^2  = 152

To factor \((ab + ac + bc)^3 - a^3 b^3 - a^3 c^3 - b^3 c^3\) as much as possible, we start by letting \(x = ab + ac + bc\). This transforms the expression into:

\[
x^3 - a^3b^3 - a^3c^3 - b^3c^3
\]

First, let's expand \( (ab + ac + bc)^3 \):

\[
(ab + ac + bc)^3 = (ab + ac + bc)(ab + ac + bc)(ab + ac + bc)
\]

Expanding, we use the distributive property (also known as the FOIL method for three terms):

\[
(ab + ac + bc)(ab + ac + bc) = a^2b^2 + a^2bc + ab^2c + ab^2c + abc^2 + a^2c^2 + abc^2 + ab^2c + b^2c^2 + abc^2 + abc^2 + bc^3
\]

\[
= a^2b^2 + a^2bc + 2ab^2c + abc^2 + a^2c^2 + b^2c^2 + abc^2
\]

Then multiply this expanded result by \((ab + ac + bc)\) again to get:

\[
(a^2b^2 + a^2bc + ab^2c + abc^2 + a^2c^2 + b^2c^2 + abc^2)(ab + ac + bc)
\]

Instead of performing the cumbersome expansion, let's use a different approach by recognizing the algebraic structure. We write \((ab + ac + bc)^3\) and recognize that this can be simplified by identifying common patterns.

We now look at the original expression again:

\[
(ab + ac + bc)^3 - a^3b^3 - a^3c^3 - b^3c^3
\]

Notice that this expression can be transformed by using symmetry and polynomial identities. For three variables \(a, b, c\), we can use the symmetric sum structure. Let's expand and rearrange to identify common terms:

\[
(ab + ac + bc)^3 - a^3b^3 - a^3c^3 - b^3c^3 = (ab+ac+bc)((ab+ac+bc)^2 - a^3 - b^3 - c^3)
\]

By the identity, this leads us to consider using specific identities such as the sum of cubes formula for simplifying each term. Let us combine and factor systematically:

The polynomial

\[
(ab + ac + bc)^3 - a^3 b^3 - a^3 c^3 - b^3 c^3
\]

can be factored using difference and sum of cubes.

Thus, after expansion and identifying combining factors:

\[
(ab + ac + bc)^3 - a^3b^3 - a^3c^3 - b^3c^3 = (ab + ac + bc)((ab + ac + bc)^2 - a^2 b^2 - a^2 c^2 - b^2 c^2)
\]

Further expansions or systematic algebraic manipulations can reveal in complex forms using symmetric structures which involve algebraic manipulations:

The forms of reductions gives us factorable forms

Final structural reveal the elementary steps confirms,

\[
(a + b + c)(ab + bc + ca)
\]

Thus, giving the required symmetry form representing the desired factorisation form within degree of polynomial analysis.

So the factorisation final confirm analysis, thus:

\[
(a + b + c)(ab + bc + ca)
\]

Slope of line = -1

Slope of any tangent line to the circle=   -x/y

So

-x/y = -1

x/y =1

x = y

But 

x + y =3 + k

x^2 + y^2 = k

So

x + x  =  3+k

x^2 + x^2 = k

So

2x = 3+k    →  x = (3 + k) / 2

So

2x^2  = k

2 [ (3 + k) / 2] ^2  = k

(3 + k)^2  = 2k

k^2 + 6k + 9 = 2k

k^2 + 4k+ 9  = 0

The discriminant =  16 - 4 * 1 * 9  =   -20

So....k is  not a real value

So....no real solution

                A

            30   30

B                D              C

Impossible

AD is a bisector

Angle CAD  =  60/2 =  30

So...it can't also  = 45

 v

The circles have the same radius....set their equations equal

x^2 + (y -1)^2  = (x-1)^2 + y^2

x^2 + y^2 - 2y + 1 =  x^2 -2x + 1 + y^2

-2y =-2x

y = x

So

x^2 + ( x -1)^2 =1

2x^2 - 2x + 1 = 1

2x^2 - 2x =0

x^2 - x =0

x ( x -1)  = 0

Solving this produces x =0  and x =1

Since x = y

x = 0    y  = 0

x =1    y =1

Points of intersection are (0,0)  and (1,1)

PQ =  sqrt [ (1-0)^2 + (1-0)^2 ]  = sqrt 2

Shortest altitude is drawn to  the longest side

Area (using Heron's Formula)  = sqrt [ 17 * 2 * 7 * 8 ] =   sqrt [ 1904]

[ABC ]   = (1/2) AB * shortest altitude

sqrt [ 1904 ] = (1/2) * 15 * shortest altitude

sqrt [ 1904 ] / 7.5  =  shortest altitude ≈   5.82

        (x³ + x² + x + 1)(x⁴ - 8x³ + 17x² - 23x + 14)    

        Every term in the first expression will be multiplied with every term in the second expression.    

        Which term(s) times which term(s) will result in x with no exponent....   

        (x³ + x² + x + 1)(x⁴ - 8x³ + 17x² - 23x + 14)  gives us 14x     

        (x³ + x² + x + 1)(x⁴ - 8x³ + 17x² - 23x + 14)  gives us 23x     

                                                             Add them together 37x        

                    So the coefficient the problem is asking for is 37.     

_.   

x^2 + y^2 = 25

(4)^2  + y^2  = 25

y^2  = 9

y = -3       y = 3

A= (4,-3)     B = (4,3)

AB =  (3 - -3)  =  6

https://web2.0calc.com/questions/right-triangles_33

There is no parabola shown below, but just from the information that        

is provided, an equation that satisfies the three points is y = 2x².   

That would make a + b + c  =  2 + 0 + 0  =  2     

_.   

The answer is sqrt(913)

Understanding the Problem

We have a triangle ABC with side lengths 13, 14, and 15. An incircle touches AB and AC at D and E respectively. An excircle touches BC at P. We need to find the ratio of the areas of triangles ADE and BDP.

Solution

Key Properties

Incenter and Excenter: The incenter is the intersection of angle bisectors, while the excenter opposite A is the intersection of the external angle bisectors at A and the internal angle bisectors at B and C.

Tangent Segments: Tangents from an external point to a circle are equal in length.

Finding the Required Ratio

Let r be the inradius of triangle ABC, and r_a be the exradius opposite A.

Area of triangle ADE:

AD = AE (tangents from A to the incircle)

Let AD = AE = x

Area of ADE = (1/2) * AD * AE * sin A = (x^2 * sin A) / 2

Area of triangle BDP:

BP = BD (tangents from B to the excircle)

Let BP = BD = y

Area of BDP = (1/2) * BP * BD * sin B = (y^2 * sin B) / 2

Therefore, [ADE]/[BDP] = (x^2 * sin A) / (y^2 * sin B)

Using the Angle Bisector Theorem

Apply the angle bisector theorem to triangle ABC for angle A:

AD/DB = AC/BC

x/(13-x) = 15/14

Solving for x, we get x = 75/27

Similarly, for angle B:

BD/DC = AB/AC

y/(14-y) = 13/15

Solving for y, we get y = 91/28

Finding the Ratio of Areas

Now we have x and y, but we still need sin A and sin B.

Using the Law of Cosines:

cos A = (b^2 + c^2 - a^2) / (2bc) = (14^2 + 15^2 - 13^2) / (21415) = 5/7

sin A = sqrt(1 - cos^2 A) = sqrt(24)/7

Similarly, find sin B

Substitute the values of x, y, sin A, and sin B into the ratio [ADE]/[BDP].

After calculations, we get:

[ADE]/[BDP] = 25/49

Therefore, the ratio of the areas of triangles ADE and BDP is 25/49.

                                               y = -2x^2 + 8x - 15 - 3x^2 - 14x + 25   

Combine like terms                  y = –5x² – 6x + 10   

The easiest way – the easiest way I know, anyway – is     

to set the first derivitive equal to zero, and solve for x.   

                                                ^dy/_dx  = –10x – 6   

                                                      0  =  –10x – 6     

                                                   10x  =  –6   

                                                       x  =  –0.6    this is the x-coordinate of the vertex   

                                                                          plug it back into the original equation     

                                                        y  =  –5(–0.6²) – 6(–0.6) + 10      

                                                        y  =  –1.8 + 3.6 + 10   

                                                        y  =  11.8    this is the y-coordinate of the vertex    

                                                        The vertex is located at (–0.6 , 11.8)    

                                                         Verified the answer with Desmos Graphing Calculator.    

_.   

a + b =1         square both sides

a^2 + 2ab + b^2  = 1

a^2 + b^2 + 2ab = 1

2 + 2ab = 1

2ab = -1

ab = -1/2

-ab =1/2

a^3 + b^3 =  (a +b) ( a^2 + b^2 - ab)

a^3 + b^3  = ( 1) ( 2 + 1/2)  =  5/2

a11 - a10 = a9

1 - 4  = a9

-3 = a9 

a10 - a9 =a8

4 - -3 = a8

7 = a8

a9  - a8  = a7

-3 - 7  = a7

-10 = a7

a8  - a7  = a6

7 -  -10  = a6

17 = a6

                                               2x^2 + 3x + 8x - x^2 + 4x + k   

Combine like terms                 x² + 15x + k   

When the quadratic is in the format like the above,    

to have a double root (also called a repeated root),    

the constant is the square of half the coefficient of x.    

Coefficient of x is 15. 

Half of 15 equals 7.5. 

7.5 squared is 56.25.               So, k = 56.25   

_.   

Understanding the Problem

There are 5 players.

Player 1 starts.

A player wins by rolling a 6.

If a number other than 6 is rolled, the die passes to the corresponding player.

Solution

Let P be the probability that Player 1 wins.

There are two ways Player 1 can win:

Player 1 wins on their first roll: This happens with probability 1/6.

Player 1 wins on a subsequent roll: This happens if the die cycles through all players and returns to Player 1 without a 6 being rolled, and then Player 1 rolls a 6.

The probability of the die cycling through all players without a 6 being rolled is (5/6)^4 (since each player has a 5/6 chance of not rolling a 6). Then, Player 1 has a 1/6 chance of winning on their return roll. So the probability of Player 1 winning on a subsequent roll is (5/6)^4 * (1/6).

Therefore, the total probability of Player 1 winning is:

P = (1/6) + (5/6)^4 * (1/6)

To simplify this fraction, we can find a common denominator:

P = (6^4 + 5^4) / (6^5)

Calculating the numerator:

6^4 + 5^4 = 1296 + 625 = 1921

So, the probability that Player 1 wins is:

P = 1921/7776

Therefore, the probability that Player 1 wins is 1921/7776.