Master Algebra 1 Word Problems with our Comprehensive Practice

Why isn't $y=a(x^2-<!--\; -\; -->Sx+P)$ same with $x^2-<!--\; -\; -->Sx+P$
If we have roots of the function $y=a{x}^2+bx+c$ we can calculate $S=\frac{-<!--\; -\; -->b}{a}$ and also $P=\frac{c}{a}$. Then we know that we can form the function this way:
$$x^2-<!--\; -\; -->Sx+P$$
So on the other side we know that we have the function $f(x)=y$ in different ways:
$$y=a{x}^2+bx+c$$
($\mathrm{\alpha <!--\; \alpha \; -->}$ and $\mathrm{\beta <!--\; \beta \; -->}$ are roots of the quadratic function)
$y=a(x-<!--\; -\; -->\mathrm{\alpha <!--\; \alpha \; -->})(x-<!--\; -\; -->\mathrm{\beta <!--\; \beta \; -->})$
And my question is here:
$$y=a(x^2-<!--\; -\; -->Sx+P)$$
Actually know that how we can form the qudratic equation using $x^2-<!--\; -\; -->Sx+P$, but the function must be like $y=a(x^2-<!--\; -\; -->Sx+P)$. Actually I don't know that why we add a. I know it will be removed when $(a)(\frac{-<!--\; -\; -->b}{a})$. But I don't know that what is $y=a(x^2-<!--\; -\; -->Sx+P)$ different whitout a!

Let $$f(x)=\frac{x+4}{x+6}$$
Find $$f^{-<!--\; -\; -->1}(-<!--\; -\; -->5)=?$$

Identify 19th term of a geometric sequence where a1 = 14 and a9 = 358.80

Write the missing term in the geometric sequence 3/4, __, 3

Find the 6th term of this geometric sequence 1, 7, 49, 343, …

For what positive value of p will this be a geometric sequence: p-3, p+1, 3p+3

Find the slope of the line perpendicular to y=−2x+3

Find ${\displaystyle a_6}$ for the geometric sequence 540,90,15,...

Add 3 odd numbers and get an even number

Find the equation of a line that passes through the point (10, 5) and is perpendicular to the line whose equation is y=54x−2

Find the indicated term of the geometric sequence where ${\displaystyle a_1=\frac{1}{3}}$, r=3, n=8

Prove that the following a geometric sequence 1, 5, -25, 125,…

Degree of Equations
A) Which variables in the formula $V=\mathrm{\pi <!--\; \pi \; -->}{r}^{2}h$ would you need to set as a constant in order to generate:
1. a linear equation?
2. a quadratic equation?
B) How should r and h be related to generate a cubic function?

Calculating endpoint for a function so arc length equals l.
I'm trying to simulate a line hanging from a given point using a quadratic function.
The line is located at point $(x_0,y(x_0))$ where y is my quadratic function.
Now, the line has length l, and I simulate it hanging and swinging from side to side by changing the quadratic and linear coefficient. This isn't really important in the problem, but I mention it just to give You a full picture.
Now, the problem is, that when I change the coefficients of the formula, the length of the line changes as well, and I have to pick a point $x_1$ in which the line will end so the length can stay the same.
Of course, the simplest way is just to pick a point $x_1$ such that distance from $(x_0,y(x_0))$ to $(x_1,y(x_1))$ is equal to l, but this actually calculates the straight-line distance, not the arc length, so for certain coefficeints this becomes really inacurate.
So, the proper way to find $x_2$ would be solving this equation:
$${\int <!--\; \int \; -->}_{x_0}^{x_1}\sqrt{1+f^{\prime }(t{)}^2}dt=l$$
For a proper $x_1$. However, after integrating it's really complicated to "extract" $x_1$ from the equation.
Thinking about this, I have found a stupid idea that doesn't work, but I dont know why, and this is the main point of my question.
So, for this integral exists a antiderivative F(x), so this whole equation can be represented as:
$$F(x_1)-<!--\; -\; -->F(x_0)=l$$
So:
$$F(x_1)=l+F(x_0)$$
Now, l is a constant, and $x_0$ is set (I'm just solving for $x_1$), so $F(x_0)$ is a constant as well. This means, that I just can differentiate the whole equation over $x_1$ and get:
$$F^{\prime }(x_1)=0=>\sqrt{1+f^{\prime }(x{)}^2}=0$$
Which is obviously wrong, because it doesn't depend at all on values of l and $x_1$.
So now, I have two questions: 1. Why is my differentiaton step wrong? 2. Is there any way to simplify solving for this integral?

Give a real life situation problem, which is mathematically translated into
2x+y+2z=18,
x+3y+3z=24,
3y=6
Please explain how this linear system models your problem.

Find the slope parallel to −25+8x=5y

Find the smallest integer if the sum of three consecutive integers is 162

A square has an area of 0.09 square inches. Find the perimeter of the square

Find the next three terms of the arithmetic sequence −13,−11,−9,−7,...

Find the formula for ${\displaystyle a_n}$ for the arithmetic sequence ${\displaystyle a_1=5,a_4=15}$