The document contains step-by-step explanations and solutions for various math problems including completing the square, solving rational equations, expanding equations, finding domains, long division, and factoring polynomials. The problems provide a demonstration of the author's mathematical understanding and were chosen to be challenging but solvable, helping the author review for an upcoming exam. The author found value in creating their own math problems and solutions through this project as it strengthened their grasp of the concepts.

1. By: Austin and Marisol

2. Completing The
Square
Step 1: You always have to move
the C to the other side by
subtracting the 215.
Step 2: Write out the equation
which is -107=6x²+26x.
Step 3: Take the 26 and ÷ it by 2 and
then square which = 169, then take
the 169 and x it by 6 which =
1014, then – 1014 by -107.
Step 4: Take the 26 and ÷ by 2
which = 13, Factor x²+26x+169
which will = (x+13)(x+13) or
(x+13)², then rewrite your equation
907= 6(x+13)², then ÷ both sides by
6.
Step 5: Write out the equation
√(907/6) = √(x+13)².
Step 6: - the 13 from x+13 from both
sides.
Step 7: Write out the final equation
x = -13 ± √(907/6)

3. Solving Rational
Equation
Step 1: Factor the Denominators to
= (x-6)(x-6), then for the 2 ÷ x-6
make the denominator the same by
x the top and bottom by (x-6)
which =
4x²+10x-24 + 2x-12 =
4
(x-6)(x-6) (x-6)(x-6) (x-6)(x-6)
Step 2: Write out the equation
4x²+10x-24 + 2x-12 =
4
(x-6)(x-6) (x-6)(x-6) (x-6)(x-6)
Then put brackets around the
equation and x it by the common
denominator which is (x-6)(x-6)
which = 4x²+10x-24+2x-12= 4.
Step 3: Then write out the equation
4x²+10x-24+2x-12= 4, then combine
like functions to simplify the
equation which = 4x²+12x36=4, then make it equal to 0 and
solve for x.
Step 4: - the 4 to make the equation
= 4x²+12x-40=0, then ÷ out the 4 on
the equation to get 4(x²+3x-10)=0.
Step 5: Factor the x²+3x-10 to get
4(x+5)(x-2).
Step 6: Then the final answer is x=5 and x=2.

4. Expanding the
Equation
Step 1: Factor the first two =
(4x-6)(4x-6) which will =
(16x²-48x+36).
Step 2: Then factor (16x²-48x+36) by (4x-6)
which =
(64x³-288x²+432x-216).
Step 3: Then factor
(64x³-288x²+432x-216) by (4x-6) which =
(256x-1536x³-3456x²-3456x+1296).
Step 4: Then factor
(256x-1536x³-3456x²-3456x+1296) by
(4x-6) which =
(1024x-7680x+23040x³-34560x²+2590x-7776).
Step 5: Write out the equation
(1024x-7680x+23040x³-34560x²+2590x-7776).
Step 6: Find the simplified form by ÷
everything by 32 which =
32(32x-240x+720x³-1080x²+810x-243).

5. Finding the Domain
Step 1: Write the equation
√(2x³+46x²-6x-138), then group
them to make them in factored
form which looks like
√[(2x³+46x²)-(6x-138)]
Step 2: Write out the equation
√[(2x³+46x²)-(6x-138)], then find the
greatest common factor in each set
which is 2x² and 6 which =
2x²(x+23) – 6(x+23).
Step 3: Notice the x+23 is in there
twice you can cancel one out and
then put the 2x² and the -6 together
which makes the equation look like
this (x+23)(2x²-6).
Step 4: Write the equation
(x+23)(2x²-6), then take the (2x²-6)
and factor that to the most
simplified form by making it = to
0, then + 6 to both sides and then ÷
both sides by 2 which = x²=3.

6. Finding the Domain
Pg. 2
Step 5: Write out the equation
x²= 3, then √ both sides which looks
like √(x²) = √(3) which = x = √(3).
Step 6: This orientation of the graph
indicates that the equation has 3
factors and has a positive A value.
The graph goes through -23, √(3), and √(3) which makes the
domain [-23, -√(3)] U [√(3), ∞).

7. Long Division
Step 1: Write out the equation
x⁶+0x-29x+0x³+244x²+0x-576, then long it
by x+3, first the x by x⁶ which makes it x
which you put on top, then you x the x with
the x which makes x⁶ and x the 3 with x
which = 3x and put that with the x⁶ then –
(x⁶+3x) from (x⁶+0x) which = -3x, then
drop down the next number to make it -3x29x.
Step 2: Continue by long the equation with
-3x-29x which would = -3x-9x, then – both
equations which = -20x+0x³.
Step 3: Continue by long the equation with
-20x+0x³ which would = -20x-60x³, then –
both equations which = 60x³+244x².
Step 4: Continue by long the equation with
60x³+244x² which would = 60x³+180x², then
– both equations which = 64x²+0x.
Step 5: Continue by long the equation with
64x²+0x which would = 64x²+192x, then –
both equations which = -192x-576.
Step 6: Continue by long the equation with
-192x-576 which would = -192x-576, then –
both equations which = 0, then write out the
equation on top with the x+3 which looks
like
(x+3)(x-3x-20x³+60x²+64x-192).

8. Long Division
Pg. 2
Step 1: Write out the equation
x-3x-20x³+60x²+64x-192, then long ÷ it by
x+2, first the x by x which makes it x
which you put on top, then you x the x with
the x which makes x and x the 2 with x
which = 2x and put that with the x then –
(x+2x) from (x-3x) which = -5x, then drop
down the next number to make it -5x-20x³.
Step 2: Continue by long the equation with
-5x-20x³ which would = -5x-10x³, then – both
equations which = -10x³+60x².
Step 3: Continue by long the equation with
-10x³+60x² which would = -10x³-20x², then –
both equations which = 80x²+64x.
Step 4: Continue by long the equation with
80x²+64x which would = 80x²+160x, then –
both equations which = -96x-192.
Step 5: Continue by long the equation with
-96x-192 which would = -96x-192, then – both
equations which = 0, then write out the
equation on top with the x+3 from the first
long and x+2 from this long which looks
like (x+3)(x+2)(x-5x³-10x²+80x-96).

9. Long Division
Pg. 3
Step 1: Write out the equation x-5x³-10x²+80x-96, then
long ÷ it by x-3, first the x by x which makes it x³
which you put on top, then you x the x with the x³
which makes x and x the -3 with x³ which = -3x³ and
put that with the x then – (x-3x³) from (x-5x³) which =
-2x³, then drop down the next number to make it
-2x³-10x².
Step 2: Continue by long the equation with -2x³-10x²
which would = -2x³+6x², then – both equations which =
-16x²+80x.
Step 3: Continue by long the equation with -16x²+80x
which would = -16x²+48x, then – both equations which
= 32x-96.
Step 4: Continue by long the equation with 32x-96
which would = 32x-96, then – both equations which = 0.
Step 5: Write out the x+3 and x+2 from the previous
long ’s and the x-3 from this one including the equation
on top which looks like (x+3)(x+2)(x-3)(x³-2x²-16x+32).
Step 6: Group the equation (x³-2x²-16x+32) which would
be x²(x-2)- 16(x+2), then you take the x-2 and join them
and take the x² and the -16 and put them together.

10. Long Division
Pg. 4
Step 7: Put all the factors together
which looks like (x+3)(x+2)(x-3)(x2)(x+4)(x²-16), then factor the x²-16
which will = (x+4)(x-4).
Step 8: Put all the factors together
which looks like (x+3)(x+2)(x-3)(x2)(x+4)(x-4).
Step 9: Solve all the factors which =
x=-3, x=-2, x=3, x=2, x=-4, and x=4.

11. The reason I chose these concepts were to make people think about how to do
them because I made them a little difficult so they can get a better grasp on
how to solve them. These problems provided a good overview of my
mathematical understanding because it showed how I solved complex
problems and were able to explain them. I learned a lot from this assignment
it helped me back track a lot so I was able to make my own equations and
solve them different ways. Yes it was valuable to me because it helped me
review for the exam coming up in the next couple of days. I thought this was
a good project to do even thought it took a long time to do.

12. I tried picking units that I can easily understand but also make it somewhat
hard. My mathematical understanding of what we learned has gotten better
from doing this project. Those problems expanded my range of the units
we’ve been learning. I did learn something from this project like long
division and grouping and finding the domain after wards. Over all the
project was a great review and it is a great idea we are doing this project.