Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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2answers
1k views

Solving Quartic Equation

Could someone please explain how to solve this : $x^4+3x^3-6x^2+16x+56=0$ - not the answer only, but a step-by-step solution.
2
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3answers
66 views

how can I solve this kind of equation?

I've got a system of equations which is: $\begin{cases} x=2y+1\\xy=10\end{cases}$ I have gone into this: $x=\dfrac {10}y$. How can I find the $x$ and $y$?
1
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1answer
76 views

How do I get the equation for this parabola in standard form?

How do I get the equation for this parabola in standard form? $ y = f(x)= 2x^2+3x-2$
0
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1answer
47 views

How do I solve this function and find its domain?

Suppose that $f(x)= -x^2+1$ and $g(x)= \sqrt{x}$. How do we find $f \circ g$ and $g \circ f$ and their domains?
1
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2answers
36 views

Any hints on how to graph this piece wise function?

I am supposed to sketch the graph of y= |x-1| if 0≤x≤2 |x-3| if 2≤x≤4 and specify any x or y intercepts i'm just confused about graphing it because of the absolute value signs. Any help or ideas?
1
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1answer
3k views

How to find the inverse of a piecewise function

How would I go about finding the inverse of a piecewise function? In this case the function is $$ f(x) = \left\{ \begin{array}{lr} x, & \text{if } 0\leq x \leq 1,\\ x-1, ...
1
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1answer
12 views

Finding combinations from a set of options?

Say I have this scenario: 240 shirts 340 pairs of shoes 90 pairs of pants 300 jackets 50 watches How can I find out the amount of combinations I can make out of these items? Would I just multiply ...
3
votes
1answer
350 views

How find this value of $x,y$

let $x,y\in R$, such $$\begin{cases} \sqrt{1+(x+y)^2}=-y^6+2x^2y^3+4x^4\\ \sqrt{2x^2y^2-x^4y^4}\ge 4x^2y^3+5x^3 \end{cases}$$ find the value of $x,y$. My try: since ...
1
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2answers
69 views

Solving $\frac{5}{t-3}-2=\frac{30}{t^2-9}$

I need help with solving this equation: $$ \frac{5}{t-3}-2=\frac{30}{t^2-9} $$ I tried to solve, but I always get false result. The result should be $-\frac{1}{2}$ but I always get $-\frac{1}{2}$ and ...
0
votes
1answer
56 views

Solving inequality. Did I do it right?

Solve the following inequality: $$5(y-2)-3(y+4)\ge2y-20$$ I made calclations and I found: $$ 0\ge2 $$ What does it mean? Are my calculations right? Here's how I did it: $$ ...
0
votes
1answer
178 views

Solve $a=x^n$ , $b=(x+1)^n$ for $x,n$

$$a=x^n~,~b=(x+1)^n$$ Just trying to solve these for $x$ and $n$ . For some reason WolframAlpha gives me a blank screen? Much thanks for any help.
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3answers
212 views

How to find the point on a parabola where x and y are equal?

On a parabola how could i find the point at which the y and x points are equal and meet on a point of the graph, algebraically?
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2answers
62 views

Find the domain and range of $f(x) = y$

$ y = \sec^{-1}(2x - x^2)$. If you graph the equation the domain and range can be inferred I don't how to solve it algebraically.
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3answers
273 views

$\log x =Cx^4$ has only one root. Find C

$\log x =Cx^4$ has only one root. Find C. I don't know how to solve this problem. Do you take derivative on both sides? I am thinking C equals 0. Am I correct on that?
0
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3answers
47 views

Integer values of $m$ for which $x^2-(m-3)x+m = 0$ has greater then $2$

Integer values of $m$ for which both the roots of the equation $x^2-(m-3)x+m = 0$ has greater then $2$ $\underline{\bf{My\;\; Try}}:$ Let $\alpha\;\;,\beta>2$ be the roots of the equation. and ...
12
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4answers
625 views

Prove maximum value of $(z-xy)(x-yz)(y-zx)$ is $\frac{1}{64}$ given $x,y,z \in (0,1)$

Prove maximum value of $(z-xy)(x-yz)(y-zx)$ is $\frac{1}{64}$ given $x,y,z \in (0,1)$ I can make it $\frac{1}{64}$ by setting $x,y,z = \frac{1}{2}$, but I have no idea how to show that's the maximum. ...
2
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1answer
100 views

$50^{th}$ digit from the left in the expansion of $(\sqrt{50}+7)^{50}$.

The $50^{th}$ digit from the left in the expansion of $(\sqrt{50}+7)^{50}$ after the decimal point. $\underline{\bf{My\; Try}}::$ Let $\left(\sqrt{50}+7\right)^{50} = I+f$, where $I = $Integer part ...
0
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1answer
472 views

How to Solve an equation with mod for a variable?

I have following equation to be solved, but I am having some trouble in making an understanding and doing so. (d * e) % v = 1 e and v are known. How to solve this ...
1
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1answer
57 views

Perpendicular line intersection issues

Do not downvote questions for being 'simple' to you. What one might find trivial another may find helpful. It is not in the spirit of SE. That being said,... I have a line with the equation $y = ...
0
votes
2answers
529 views

How do I solve this exponential function? $2^{-100x} = (0.5)^{x-4}$

How do I solve for $x$? $2^{-100x} = (0.5)^{x-4}$
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3answers
86 views

Problem finding in simple algebra

It is given, $$x= \sqrt{3}+\sqrt{2}$$ How to find out the value of $$x^4-\frac{1}{x^4}$$/ The answer is given $40 \sqrt{6}$ but my answer was not in a square-root form I have done in thsi way: $$x+ ...
0
votes
2answers
45 views

Can't seem to figure out steps of factoring

I can't seem to figure out how to get from $5(5\cdot 3^{n}-3\cdot 2^{n})-6(5\cdot3^{n-1}-3\cdot2^{n-1})$ to $(5\cdot5\cdot3-6\cdot5)3^{n-1}-(5\cdot3\cdot2-6\cdot3)2^{n-1}$ by factoring out ...
0
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2answers
54 views

Finding the mistake in precalc equation

$$\sqrt{x+3}+\sqrt{x-1} = -1;\tag{1}$$ $$ \mathbb{D}=[1,\infty)\tag{2}$$ $$x+3=(-1-\sqrt{x-1})^2\tag{3}$$ $$x+3=1+2\sqrt{x-1}+x-1\tag{4}$$ $$x_{1/2} = \pm\sqrt{\frac 3 2}+1\tag{5}$$ $$\mathbb{D} -> ...
0
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2answers
51 views

Find asymptotes of $(2x)/(x-1)^2$

What are the asymptotes of $$\frac{2x}{(x-1)^2}$$ ? I have problems already on domain.
1
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1answer
193 views

Logarithm Problem : Find the number of real solutions of the equation $2\log_2\log_2x+\log_{\frac{1}{2}}\log_2(2\sqrt{2}x)=1$

Find the number of real solutions of the equation $2\log_2\log_2x+\log_{\frac{1}{2}}\log_2(2\sqrt{2}x)=1$ My approach : Solution : Here right hand side is constant term so convert it into log ...
8
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0answers
275 views

Is there an elementary introduction to higher order functions?

I am teaching a pre-calculus course (using the textbook by Michael Sullivan if it helps), and I realized that higher order functions seem to show up in with some frequency in pre-calculus and ...
2
votes
2answers
84 views

Question on interpretation of Dot Product

Everyone says that the dot product is interpreted as the projection of A onto B (if you are dot producting A and B), but isn't that length just equal to |A|$\cos \left( \theta \right)$? Why does the ...
2
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2answers
2k views

Solving system of equations with R

Suppose I have the following function : $f(x,y,z,a)= \cos(ax) + 12y^2 - 9az$ and I want to solve the following syste of equations. $ f(x,y,z,1)= 10 $, $f(x,y,z,5)= 7 $, and $f(x,y,z,-3)= 17 ...
3
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1answer
42k views

Finding Revenue Function and Max Revenue

Studying for a midterm. The demand function for a manufacture's product is $p=1000-\frac1{80} q$ Where $p$ is the price (in dollars) per unit when $q$ units are demanded (per week) by consumers. ...
0
votes
1answer
37 views

Inequality and absolute value: $p + |k| \gt |p| + k$

Here is the problem I am confused about. The given relation is: $p + |k| \gt |p| + k$ It is not mentioned whether $p$ and $k$ are integers. I need to determine whether $p$ and $k$ are ...
11
votes
3answers
274 views

Prove that x is rational

Let $x$ be a real number with the properties that $x^3+x$ and $x^5+x$ are rational. Prove that $x$ is rational. Denote $a=x^3+x$; $b=x^5+x$. We can multiply and add them together until we get the ...
0
votes
1answer
71 views

Solving for a variable with square roots in equation

I'm working on the following equation: $$ a= \frac{2}{t^2}[(d+l)^{1/2} - (d^{1/2})]^2 $$ I want to solve for l such that: $$ l=\frac{1}{2}at^2+(\sqrt{2ad}) t $$ I can't get it in this form. Here's ...
4
votes
4answers
274 views

Why is this defined even when divided by zero

so I've got $$\dfrac{x^3-4x^2+3x}{x^2-1}$$ and want to calculate the asymptotes. There's one a $x=-1$ since the function is not defined there. But the function seems to be defined for $x=1$. How come? ...
2
votes
3answers
140 views

Inequalities with absolute value question

Solve: $ 5< \left\vert\dfrac{x+10}{x-10}\right\vert<6$ attempt at a solution: Dividing into two: $5<\left\vert \dfrac{x+10}{x-10}\right\vert $ And $\left\vert ...
2
votes
2answers
3k views

How to derive the equation of a parabola given a focus and a directrix not parallel to the x or y axis?

I was wondering if it is possible to derive a general form of a parabola given any focus and directrix. So far all the materials I have come across only show the derivation for a parabola equation ...
1
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1answer
50 views

Solving systems of linear equations

Ali and Zayd have $8.80 in pennies and nickels. The number of pennies is twice the number of nickels. Let p represent the number of pennies and n represent the number of nickels. Set up the problem ...
12
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11answers
2k views

Solve $4^{9x-4} = 3^{9x-4}$

I am having some trouble trying to solve $$4^{9x-4} = 3^{9x-4}$$ I tried to make each the same base but then I'm becoming confused as to what to do next. These are the steps I took: ...
0
votes
3answers
544 views

Represent $\frac{1}{3} \ln(x+2)^3 + \frac{1}{2}[\ln x - \ln (x^2+3x+2)^2]$ as a single logarithm

I am having some trouble trying to find the single logarithm for the following: $$\frac{1}{3} \ln(x+2)^3 + \frac{1}{2}[\ln x - \ln (x^2+3x+2)^2]$$ I understand that I have to use the addition and ...
2
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2answers
401 views

Goat tethered in a circular field

I need help on this question. I was thinking of counting the area of the whole circle and then subtracting it with the area that is not eaten by the goat. But I don't know how to find this particular ...
0
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4answers
377 views

Induction: Show that $\sin(2x) + \sin(4x) + \ldots+ \sin(2nx) = \frac{\sin(nx)\sin((n+1)x)}{\sin(x)}$

Show that $\sin(2x) + \sin(4x) + \ldots+ \sin(2nx) = \dfrac{\sin(nx)\sin((n+1)x)}{\sin(x)}$ I tried to use induction. Base case is easy, but I'm stuck at the induction step (from $k$ to $k+1$). ...
2
votes
1answer
91 views

Compound quadratic problem

The first issue I have is that I am not sure why this is called a 'compound quadratic problem', but anyway to proceed: Suppose that $x-y=14$ and $$(x+y)(x^2+y^2)(x^4+y^4)=a(x^b-y^b)$$ where $a$ and ...
0
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2answers
122 views

How to solve $x+(x\cdot\frac{20}{100})=600$?

I was solving a puzzle, and finally I got this equation as the result but I couldn't solve it. $$x+\left(x\times\left(\frac{20}{100}\right)\right)=600$$ How to solve this equation? Please ...
1
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1answer
94 views

How are solutions are there for $\cos(97x) = x$?

How are solutions are there for $\cos(97x) = x$? Could anyone please tell me how to start?
0
votes
2answers
40 views

Calculation of all values of $a$ for which $3x^2+(4-2a)x-8-a^2\leq 0$.

Calculation of all values of $a$ for which $3x^2+(4-2a)x-8-a^2\leq 0$. Given that $x$ lies between $-3$ and $2$. My Try:: Let $x = \alpha,\beta$ be the Roots of Given equation. where ...
1
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1answer
13 views

If $|z| = 2$. Then Locus of $z$ Representing The Complex no. $-1+5z,$ is

If $|z| = 2$. Then Locus of $z$ Representing The Complex no. $-1+5z,$ is $\underline{\bf{My\;\; Try::}}$ Let $z^{'} = -1+5z$, Where $z^{'} = x^{'}+iy^{'}$. So put $5z = z^{'}+1$ in $|z| = ...
4
votes
2answers
347 views

Solving system of multivariable 2nd-degree polynomials

How would you go about solving a problem such as: \begin{matrix} { x }^{ 2 }+3xy-9=0 \quad(1)\\ 2{ y }^{ 2 }-4xy+5=0 \quad(2) \end{matrix} where $(x,y)\in\mathbb{C}^{2}$. More generally, how would ...
0
votes
2answers
49 views

Logarithmic functions?

I am stuck on this question which as follows: $\log(x) + \log(x-3) = \log(10x)$ I have tried the following and not sure if I am doing it correctly... 1) $$ \log(x) + \log(x-3) = \log(10x) ...
0
votes
3answers
41 views

Using Logarithms

\begin{align*} -2^{n-1} \ln2 &= -100 \ln 10\\ &\\ -100 \ln 10 &= -230\\ &\\ \dfrac{-230}{\ln (2)} &= -333\\ &\\ -2^{n-1} &> -333\\ &\\ (n-1) \ln(-2) &> ...
2
votes
2answers
237 views

Rate of Change of Cylindrical Roll's Volume as it Unrolls

This is purely a "for-fun" question. I was minding my own business in the washroom this morning when I began to unroll some toilet paper from the roll, and in typical Breaking Bad fashion (sorry if ...
0
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1answer
53 views

Finding the positive value K

Say we have some positive value K. How can we know what K is if the function $f(x)=K tan(x)$ intersects its derivative perpendicularly on a graph? So I know how to do this using some form of a ...