Master Algebra 1 Word Problems with our Comprehensive Practice

Find an equation of the quadratic function with zeros at (0,0) and (6,0) with $f(5)=-<!--\; -\; -->15$
write the equation of the quadratic function with zeros at (0,0) and (6,0) with $f(5)=-<!--\; -\; -->15$
So, I know how to get the equation from the zeros, but I am confused with what I am supposed to do with "$f(5)=-<!--\; -\; -->15$".

How to prove the existence of a minimum of a quadratic function of two variables?
$$f(x,y)=A{x}^2+2Bxy+C{y}^2+2Dx+2Ey+F,\text{where }A>0\text{ and }{B}^2<AC.$$
Prove that a point (a,b) exists which f has a minimum.
I figured out that there is no stationary point for this equation.
So, Hessian Matrix seems not helpful.
In my book, it says that "change quadratic part to sum of squares
but, Can't think of any way to change it to sum of squares.
Also,
Why $f(a,b)=Da+Eb+F$ is at this minimum..?

What is the probability that a random number is irrational/transcendental? Are there infinitely more transcendental numbers than irrational numbers?

$e^{pt}$or $(1+p{)}^t$ What is the difference in modeling exponential growth and decay?
I would really like to better recognize, whilst the feature $e^{pt}$ is the "higher" desire and while (if at all) $(1+p{)}^t$ have to be used.
to provide a conventional example: Say we need to version radioactive decay of some detail A. allow t be in units of one half-life of A. Then $p=-<!--\; -\; -->\frac{1}{2}$
Now I'd say the standard approach to modeling this is via the function
$$A_1(t)=A_0\cdot <!--\; \cdot \; -->{(1-<!--\; -\; -->\frac{1}{2})}^t.$$
On the other hand, if we approach this problem as an ODE, we can say that at any point t we want A(t) to decrease at a rate of half of its momentary amount:
$$\frac{d}{dt}{A}_2(t)=-<!--\; -\; -->\frac{1}{2}{A}_2(t),$$
which leads to the function
$$A_2(t)=A_0\cdot <!--\; \cdot \; -->e^{-<!--\; -\; -->\frac{1}{2}t}.$$
But which approach would be "better" here? I think $A_1$ is much more commonly (if not exceptionally) used when it comes to modeling atomic decay. On the other hand, I know that
$$e=\underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}{(1+\frac{1}{n})}^n$$
which essentially means that the rate of change of $A_2$ is continously updated, while $A_1$ is updated discretely, right? That's the best way I can phrase it at the moment.
So to conclude: Does this mean that $A_2$ is always the "better", more accurate choice or are there situations where $A_1$ is actually "correct"?

Create a linear model in a word problem

If the measures of the three angles of a triangle are x, 2x-20, and 3x-10, the what is this type of triangle?

Twice the difference of a number and 8 is equal to three times the sum of the number and 3. Find the number

Solve the following system of partial differential equations
$$\begin{array}{r}\{\begin{array}{l}\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}a}S(a,b,c,d)=f_1(a)\\ \frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}b}S(a,b,c,d)=f_2(b)\\ \frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}c}S(a,b,c,d)=f_3(c)\\ \frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}d}S(a,b,c,d)=f_4(d)\end{array}\end{array}$$
where $f_i()$s are some nonlinear functions.
Does the above system have a unique answer? And if has can any one introduce a reference, explaining the techniques for analytic solutions?

Would $-<!--\; -\; -->3^{-<!--\; -\; -->x}$ be an exponential decay of growth? Any and all help appreciated.

Let $f:\mathbb{R}⟶<!--\; ⟶\; -->\mathbb{R}$ be a differentiable function. If
$$\underset{x\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}\frac{f(x)}{x}=1,$$
then there exists a sequence $(x_n)$ such that $x_n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$ as $n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$ and
$$\underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}{f}^{\prime }(x_n)=1.$$
We tried to use the Mean Value Theorem. For each n, there exists $x_n\in <!--\; \in \; -->[0,n]$ such that
$$f^{\prime }(x_n)=\frac{f(n)-<!--\; -\; -->f(0)}{n}.$$
Hence, $\underset{n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}{f}^{\prime }(x_n)=1$. But, we are not sure that $x_n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$

Write the common ratio of this geometric sequence 1.1, 4.4, 17.6, 70.4,...

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

Show that the set of irrational numbers fails to be closed

Find the slope that is perpendicular to the line y=(−3x)

Find the slope parallel to −15+3y=−12x

Tell whether the sequence 3,5.5,8,10.5,13 is arithmetic

Find the slope of any line perpendicular to the line passing through (−8,2) and (−12,−20)

Sydney, Australia is 13 hours ahead of my current location. If it's 3:00 pm here, what time is it in Sydney, Australia?

Write an equation of a line through (-5,3) parallel to ${\displaystyle y=\frac{2}{3}x+3}$

Suppose $a,b$ are two irrational numbers such that ab is rational and $a+b$ is rational. Then $a,b$ are the solution to a quadratic polynomial with integer coeffecients.