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Certainly, let's find the area of rectangle WXYZ.

1. Understand the Trisection

Since WA and WB trisect ∠ZWX, we know that ∠ZWA = ∠AWB = ∠BWX.

2. Let's Define

Let's denote the length of rectangle WXYZ as 'l' (which is also equal to YZ)

Let's denote the width of rectangle WXYZ as 'w'

3. Analyze Triangles

Consider triangles ΔWZA and ΔWZB.

These triangles share a common side (WW) and have equal angles (∠ZWA = ∠BWX).

Since the rectangle has 90-degree angles, ∠WZA = ∠WBZ = 90 degrees.

Therefore, triangles ΔWZA and ΔWZB are congruent by the Angle-Side-Angle (ASA) congruence criterion.

4. Implications of Congruence

Since the triangles are congruent, their corresponding sides are equal.This means:

WA = WB

AZ = BZ

5. Calculate ZB

We are given AZ = 4.

Since AZ = BZ, we have BZ = 4.

6. Calculate YZ

YZ = BY + BZ

YZ = 2 + 4

YZ = 6

7. Determine the Length and Width

Since YZ is the length of the rectangle, l = 6.

Since WA = WB and they trisect ∠ZWX, WA and WB bisect the rectangle.

Therefore, width (w) = AZ = BZ = 4.

8. Calculate the Area

Area of rectangle WXYZ = length * width

Area = l * w

Area = 6 * 4

Area = 24

Therefore, the area of rectangle WXYZ is 24 square units.

The answer is f(n) = 4n^3 + 5n^2 + 9n.

ab^4 = 48         (2)

ab^3  = 24        (1)

Divide (2) by (1)

b = 48 / 24 =  2

So

a (2)^4  = 48

a * 16  = 48

a = 48 / 16  = 3

(a , b)  = ( 3 , 2)

You do this with trigonometric identities.

a + b = 1      square both sides

(a + b)^2  = 1

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)

                 =  (1) ( 2 + 1/2)

                  =    5 / 2

a11 = a10 + a9

1 = 4 + a9

a9 = -3

a10  = a9 + a8

4 = -3 + a8

a8 = 7

a9 = a8 + a7

-3  = 7 + a7

-10 = a7

a8  = a7 + a6

7 = -10 + a6

a6 = 17

We need to solve these equations equating radii

x^2 + y^2  = ( x + 3)^2 + (y -2)^2

x^2 + y^2  =  x^2 + 6x + 9  + y^2 - 4y + 4

4y - 6x  =  13      (1)

And

x^2 + y^2   = (x + 2)^2  + (y -1)^2

x^2 + y^2 = x^2 + 4x + 4 + y^2 -2y + 1

2y - 4x  = 5    multiply this through by -2  →   -4y + 8x = -10     ( 2)

Add (1) , (2)

2x  =  3

x  = 3/2

And 

2y - 4(3/2)  = 5

2y - 6  = 5

2y = 11

y = 11/2

Center  =  ( 3/2 , 11/2)

https://web2.0calc.com/questions/algebra_53554#r1

https://web2.0calc.com/questions/help-geometry_57877

https://web2.0calc.com/questions/geometry_47534

I'm not sure if you can use 1 in the first blank but here's what I got

\(3x^2+12x-4-2x^2+6x+7 = x^2+18x+3 =(x+9)^2-78 =1(x+9)^2+(-78)\)

15 * 45  =   20 * x

675  = 20x

675 / 20  =  x =  33.75 min

Since it passes through (0,0) then c  =  0

So

a(-2)^2 + b(-2)   =  8

a(2)^2 + b(2)  =  8           add these

8a   =  16

a =  2

And to find b

2 (2)^2 + b(2)  = 8

8 + 2b = 8

2b = 0

b  = 0

a + b + c  =

2 + 0 + 0  =  

2

3x^2 + 12x - 4 - 2x^2 + 6x + 7 = ___ (x + ___ )^2 + ___
x^2+18x+3=  ___ (x + ___ )^2 + 3
x^2+18x=___(x+___)+ ____
(x+9(cuz 9*2=18)--->  (x+9)^2= x^2+18x+81
so answer is 1(x+9)^2+84

Find the vertex of the graph of the equation y = -2x^2 + 8x - 15 - 3x^2 - 14x + 25.    

                                  y  =  –2x² + 8x – 15 – 3x² – 14x + 25    

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

set the first derivitive to zero    

                                   –10x – 6 = 0     ==>>     x = – 6 / 10  =  – 0.6    

plug this x back into the equation    

                                        y  =  –5•(–6/10)² – 6•(–6/10) + 10    

                                        y  =  –180 / 100  +  36 / 10  +  10    

                                        y  =  –180 / 100  +  360 / 100  +  1000 / 100    

                                        y  =  1180 / 100  =  59 / 5  =  11.8    

Thus, the vertex is at (–0.6, 11.8)    

_.    

At a cafeteria, Mary orders two pieces of toast and a bagel, which comes out to $3.15. Gary orders a bagel and a muffin, which comes out to $3.50. Larry orders a piece of toast, two bagels, and three muffins, which comes out to $8.15. How many cents does one bagel cost?    

                                        T + B = 315 ¢    ==>>    T = 315 – B    

                                        B + M = 350 ¢    ==>>    M = 350 – B    

                                         T + 2B + 3M = 815 ¢   

                                         (315 – B) + 2B + [3 • (350 – B)]  =  815    

                                         315 – B + 2B + 1050 – 3B  =  815    

                                          – 2B + 1365  =  815    

                                          – 2B  =  – 550    

                                               B  =  275 ¢   

_.    

(x+2)/(x) • (x+3)/(x+1) • (x+4)/(x+2) = 10    

cancel the (x+2)    

(1)/(x) • (x+3)/(x+1) • (x+4)/(1) = 10    

[ (1) • (x+3) • (x+4) ] / (x) • (x+1) • (1) ] = 10    

(x² + 7x + 12) / (x² + x) = 10    

x² + 7x + 12 = 10x² + 10x    

0 = 9x² + 3x – 12     this will factor    

(3x + 4) (3x – 3) = 0    

                      3x + 4 = 0     x will not be an integer, so disregard    

                      3x – 3 = 0    

                            3x = 3    

                              x = 1     therefore (x+2)/(x) = 3/1 = 3 is the value of the first fraction   

_.    

Find all values of $x$ such that $x^2 - 5x + 4 = -x^2 - 25x - 14$. If you find more than one value, then list your solutions, separated by commas.  

x² – 5x + 4  =  – x² – 25x – 14    

2x² + 20x + 18  =  0    

  x² + 10x + 9  =  0    

(x + 1)(x + 9)  =  0    

x = –1 and x = –9    

_.    

A pole of height 50 units has broken into two pieces, and the top of the pole is now at a point that is 25 units from the base of the pole. Find the difference between the lengths of the two pieces.    

The pole was 50 feet high.  The pole is now 25 feet high.    

That means the piece broken off is 50 – 25 = 25 feet long.    

The two pieces are the same length.  The difference is zero.    

_.    

The answer is [YBC]/[ZYC] = 9/5.