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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-1
votes
0answers
16 views

Reducing algebraic summation

I am a computer programmer by trade and am studying algorithm analysis...because i am masochistic like that. Anyhow, I was looking at the solution for one of the problems in the book. However, I am ...
3
votes
3answers
50 views

What $n^{\frac{1}{\log_2n}}$ means?

I was confused with about the $n^{\frac{1}{\log_2n}}$ expression. I am not sure how to make mathematical sense of it - i.e. express it in another way for easier understanding. I tried to plug in some ...
2
votes
4answers
70 views

$x^2+y^2+9=3(x+y)+xy$ Find all pairs of real $x,y$ that meet this equation

$\frac{(x-y)^2}{(y-3)(3-x)} = 1$ That was my attempt, I can't think of anything else here. I'd prefer a hint
3
votes
5answers
120 views

$a^2 + b^2 + c^2 = 1 ,$ then $ab + bc + ca$ gives =?

In a recent examination this question has been asked, which says: $a^2+b^2+c^2 = 1$ , then $ab + bc + ca$ gives = ? What should be the answer? I have tried the formula for $(a+b+c)^2$, but gets ...
1
vote
2answers
58 views

Evaluate $(1-\frac1{2^2})(1-\frac1{3^2})\ldots(1-\frac1{2015^2})$

Evaluate $$\prod_{k=2}^{2015} \left(1-\frac1{k^2}\right) = \left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\ldots\left(1-\frac{1}{2014^2}\right)\left(1-\frac{1}{2015^2}\right)$$
0
votes
0answers
8 views

Equations: Find $c,b,f$ if $c,b,f>0$

I am given $c^2+f^2+cf=49$, $c^2+b^2-cb=49$ and $f^2+b^2-fb=49$. Find $c,b,f$ if $c,b,f>0$ I couldn't do this by hand, please help All I can find out is that $c+f=b$.
3
votes
4answers
93 views

no. of real roots of the equation $ 1+\frac{x}{1}+\frac{x^2}{2}+\frac{x^3}{3}+…+\frac{x^7}{7} = 0$

The no. of real roots of the equation $\displaystyle 1+\frac{x}{1}+\frac{x^2}{2}+\frac{x^3}{3}+............+\frac{x^7}{7} = 0 $ $\bf{My\; Try::}$ First we will find nature of graph of function ...
1
vote
0answers
10 views

Help demonstrate how to arrive at the implication of some given inequalities and equations

Given: $0<x<y<1$ $z=x+y$ $x=u$, $y=z-u$ $0<u<z-u<1$ I need to show that this implies: $0<u<\frac{z}{2}$, if $0<z<1$, and $z-1<u<\frac{z}{2}$, if $1<z<2$ ...
1
vote
0answers
56 views

polynomial equations of degree 4

I got this polynomial equations of degree 4 $$x^{4}-6x^{3}-36x^2+216x-324=0$$ from Crossed Ladders Problem and i'm tired to solve it without using WF or any software calculator I even read solution ...
1
vote
1answer
25 views

Calculating with brackets [on hold]

I have the formula MV = -0.05 -5(log11.25-1) = -0.31 . I don't understand the 5 part, how would I put this formula into a spreadsheet (Planmaker) ? Thanks
3
votes
1answer
40 views

Proving that $\sin^7\theta + \cos^7\theta <1$ using basic trigonometry and identities [on hold]

How do I prove $\sin^7\theta + \cos^7\theta < 1$ for an angle between $(0,\pi/2)$?
2
votes
4answers
48 views

How can I square $-1 < x < 1$?

If I square $-1 < x < 1$, I get $1 < x^2 < 1$ which doesn't make any sense. What additional algebraic steps do I need to apply in order to get the proper inequality $0 < x^2 < 1$? ...
0
votes
0answers
35 views

Find L for $r = \cos 3 \theta$.

Pictured above is the graph of $r = \cos 3 \theta$ for $0 \le \theta \le L$. Find the smallest value of $L$ that still produces the entire graph of $r = \cos 3 \theta$. I am having trouble starting ...
-3
votes
0answers
24 views

Fill in the blanks. (algebra) [on hold]

Fill in the blanks. *blank*(n + 4) = 6n+24 *blank*(m - *blank*) = 6m-12 *blank*(6p + 9) = *blank*p + 81 3(*blank*q - *blank*) = 18q -6
2
votes
2answers
62 views

System of equations with 2 parameters

I have no idea how even to start! \begin{align*} (u^2+v^2)(u+v)&=15uv \\ (u^4+v^4)(u^2+v^2)&=85u^2v^2 \end{align*}
0
votes
1answer
26 views

Let $x, y,$ and $z$ be positive real numbers that satisfy $2 \log_x (2y) = 2 \log_{2x} (4z) = \log_{2x^4} (8yz) \neq 0$…

Let $x, y,$ and $z$ be positive real numbers that satisfy $2 \log_x (2y) = 2 \log_{2x} (4z) = \log_{2x^4} (8yz) \neq 0$. The value of $xy^5 z$ can be expressed in the form $\frac{1}{2^{p/q}}$, where ...
-3
votes
1answer
35 views

Why is this wrong? Simple algebra [on hold]

If i am correct how would I complete the problem from there?
0
votes
2answers
27 views

…and a and b are relatively prime positive integers. Find a+b. [on hold]

Let $P = \log_a b$, where $P = \log_2 3 \cdot \log_3 4 \cdot \log_4 5 \cdots \log_{2008} 2009$ and $a$ and $b$ are relatively prime positive integers. Find $a+b$.
0
votes
2answers
76 views

$f(xy)=\frac{f(x)+f(y)}{x+y}$ Prove that $f$ is identically equal to $0$

For all $x,y\in\mathbb{R}$. also $f : \mathbb{R} → \mathbb{R}$ and $x+y\not=0$. My attempt: I restated it as $a[x^2 y^2 (\frac{x}{y}+\frac{y}{x}-\frac{1}{y^2}-\frac{1}{x^2})] + ...
0
votes
3answers
47 views

If $f(ab)f(ac)f(bc)f(a+b)f(a+c)f(b+c)=2015$ for every positive (non-zero) $a$, $b$ and $c$, find $f(2016)$.

If $f(ab)f(ac)f(bc)f(a+b)f(a+c)f(b+c)=2015$ for every positive (non-zero) $a$, $b$ and $c$, find $f(2016)$. Can someone help me?
1
vote
2answers
45 views

Showing that $\alpha \beta$ is the root of a polynomial

Assuming that $\alpha, \beta$ are distinct roots of $P(x) = x^4+bx^3-1 = 0$, where $b \in \mathbb R$, show that $\alpha \beta$ is the root of $Q(x) = x^6+x^4+b^2x^3 -x^2 -1$. I have already noticed ...
0
votes
1answer
27 views

Find the sum of the squares of all sides and diagonals of a n-gon inscribed in a circle.

With a circle with radius r and center A, for any homogenous n-gon -- find the sum of the squares of all sides and diagonals of the n-gon inscribed within the circle. I believe the general rule for ...
0
votes
1answer
41 views

If $a^{1/x}=k$ then how is $a=k^x$? [on hold]

If $a^{1/x}=k$ then how is $a=k^x$? It's a basic thing but I'm having a little problem understanding this thing.
1
vote
0answers
29 views

Solve these simple simultaneous equations?

Assuming $x_1, x_2 \geq 0, \lambda \neq 0, w_1,w_2 > 0$ We have the equalities: $$w_1 - \lambda x_2 = 0 ... (1)$$ $$w_2 - \lambda x_1 = 0 ... (2)$$ $$\bar y - x_1x_2= 0 ... (3)$$ My solutions ...
2
votes
1answer
13 views

Solve for a hyperbolic Laplace Transform by expressing as exponents and shiftig on s-axis (5.3-21)

I cannot get past a certain point on this problem as shall be shown. I need guidance in order to complete the problem. The exercise as stated in the text: Represent the hyperbolic function in terms ...
1
vote
4answers
74 views

If $2x = a + b + c$, show that $(x − a)^2 + (x − b)^2 + (x − c)^2 + x^2 = a^2 + b^2 + c^2$ .

Having trouble solving this. If $2x = a + b + c$, show that $(x − a)^2 + (x − b)^2 + (x − c)^2 + x^2 = a^2 + b^2 + c^2$. .
0
votes
2answers
38 views

Re-arranging an equation help

How do I re-arrange the equation $$ -150 = (-9.8)(t) + 0.5(-9.8)(t)^2 $$ and solve for $t$? I collected the like terms firstly, so $$-150 = -48.02 \cdot t^3$$ then I knew I was doing something ...
2
votes
2answers
33 views

given a circle $(x-1)^{2}+ y^{2}=1$, find $b$ such that the line $y=x+b$ intersects with the circle just once.

given a circle $(x-1)^{2}+ y^{2}=1$, find $b$ such that the line $y=x+b$ intersects with the circle just once. This question is for a precalculus class so setting the derivative of the positive ...
2
votes
2answers
161 views

Solving a second-degree exponential equation with logarithms

The following equation is given: $8^{2x} + 8^{x} - 20 = 0$ The objective is to solve for $x$ in terms of the natural logarithm $ln$. I approach as follows: $\log_8{(8^{2x})} = \log_8{(-8^{x} + ...
1
vote
3answers
32 views

Problem with simultaneous equations

Given that $(5, h)$ is a solution of the simultaneous equations $h(x-y) = x + y -1 = hx^2 - 11y^2$, find (a) the value of $h$. (b) the other solution of the equation.(x and y) I don't ...
1
vote
1answer
46 views

Precalculus - connect 2 towns

A state highway department plans to construct a new road between towns $A$ and $B$. Town $A$ lies on an abandoned road that runs east-west. Town $B$ is $20$ miles north of the point on that road that ...
1
vote
1answer
33 views

How to simplify logs and powers?

Is there any way to simplify $(\log a)^{\log b} = c$? And even this $(\log x)^y = z$? And also this $(\log m)(\log n) = p$ (which is essentially $\log m^{\log n} = p$) I was trying to simplify some ...
0
votes
2answers
24 views

Find the Domain and Sketch the Graph of the Function $h(x)= \frac{3x+|x|}{x}$

\begin{align*} h(x) & =\dfrac{3x+|x|}{x}\\ & = \begin{cases} \dfrac{3x+|x|}{x} & \text{if $x > 0$}\\ \dfrac{3x + |-x|}{x} & \text{if $x < 0$} \end{cases} \end{align*} I ...
0
votes
1answer
58 views

$a,b,c$ are integers ,and $ \frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ ,$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ are integers, prove $a=b=c$

Here is my solution; can someone check it out? The sum $$ \frac{b}{a}+\frac{c}{b}+\frac{a}{c} $$ is just $$ \frac{a}{b}+\frac{b}{c}+\frac{c}{a} $$ but 'reversed'. No element of this expression can ...
1
vote
1answer
43 views

$ab-(a+b)(a-b)=0$ and the Golden ratio.

I have found: $b=a*\phi$ $b=a*(-\phi)$ $b=a/\phi$ Trying to find the correlation with the equation and phi, any insight how to demonstrate this or a proof?
1
vote
2answers
35 views

Given some of the roots of the function $f(x) = x^3+bx^2+cx+d$, how do I find the coefficients of that function?

Two of the roots of $f(x) = x^3+bx^2+cx+d$ are $3$ and $2+i$. How do I find b+c+d? The answer choices are -7, -5, 6, 9, and 25.
0
votes
0answers
24 views

What is the best book to learn coordinate geometry

The level should be above high school, and it must be free online if at all possible. Also, I have an additional question to ask: How many months, roughly, would it take to finish a mathematics book ...
0
votes
1answer
41 views

Existence of solution for an equation including polynomial and trinogometric sum

Prove that the following equation has at least a solution in $[-\pi, \pi]$ : $$ x^5+\sum^{n}_{k=1}(a_k\cos kx+b_k\sin kx)=0 $$ I think the existence of the solution on $[-\pi, \pi]$ strongly depends ...
1
vote
1answer
25 views

Maximum and minimum of a fractional function

Let $x, y \in \mathbb{R}$, $a, b, c$ are three real parameters with $c\neq 0$. Find the maximum and minimum of $\dfrac{ax+by+c}{\sqrt{x^2+y^2+1}}$ This is quite complicated if I calculate the ...
3
votes
1answer
86 views

$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$ is Irrational

If $m_1 , m_2, \cdots m_n$ are natural numbers where at least one of them is not a perfect square, then how do I prove that the sum $$\sqrt{m_1}+\sqrt{m_2}+ \cdots + \sqrt{m_n}$$ is irrational? I'm ...
0
votes
2answers
35 views

variables to the power of a fraction

I have this question for advanced math, I can't seem to get my head around. $$\frac{x^{5/2}}{(x^{1/3})^4}$$
0
votes
7answers
671 views

Is it possible to find the product of two numbers given their difference?

That is, find $a\cdot b$ given the value of $a-b$. Is it possible?
0
votes
1answer
42 views

How to solve quartic polynomial equation

Can someone tell me how to solve $x^4 + 6x^2 + 5 = 0$? I know what to do when each term has an exponent one less than the previous term (e.g., $x^4 + 3x^3 + 6x^2 + 5 = 0$), but not when exponents are ...
0
votes
0answers
13 views

Displaying a 3D function without a graph

I have a 3D function $z=\dfrac{x}{y}$, and I have no access to a function grapher, but I still need to display this function in a comprehensible way. I thought of a table, but even with a domain of ...
-1
votes
0answers
20 views

Combing piecewise functions [on hold]

How would I combine the following two piecewise functions in terms of addition and subtraction? How would I find $f(x) + g(x)$, and also $f(x) - g(x)$? Thanks!
0
votes
0answers
21 views

solve this equation $1 = \left( \frac{\mu}{f} \right)^{\frac{3}{2}} \left( 1+ \frac{ \pi^2}{8} \left( \frac{kT}{\mu} \right)^2 \right)$

I am supposed to solve $1 = \left( \frac{\mu}{f} \right)^{\frac{3}{2}} \left( 1+ \frac{ \pi^2}{8} \left( \frac{kT}{\mu} \right)^2 \right)$ iteratively for $\mu$ and am supposed to get $$\mu = f ...
20
votes
7answers
2k views

How to make a “function”?

I dropped out of school early when I was still a teenager and now I'm trying to take my GED. I'm really close to passing but I'm still having trouble understanding some concepts. In the pre-test, ...
0
votes
1answer
16 views

Exclude one function from another

Is it possible to find a function, $g(n)$ that will include all the natural values except those in $f(n)$? $$f(n) = 3n$$ $$g(n) = 1,2,4,5,7,8,10...$$
0
votes
2answers
25 views

Line Intersect in Diagonals of a Rectangle

The diagonals of the rectangle have these equations: $$y = 4x-10\\ \\ y = -4x+18$$ Find the point at which the diagonals intersect. First, I tried working out $(x,y)$ $4x - 10 = -4x + 18$ $4x = ...
0
votes
0answers
8 views

Bound all $k$-th derivatives by directional derivatives of order $k$

Assume $f\in C^k(\mathbb{R}^n)$, $x\in\mathbb{R}^n$, and $|(\partial_\xi)^kf(x)|\leq 1$ for all $\|\xi\|=1$. Which bounds do we have for $|\partial^\alpha f(x)|$ when $|\alpha|=k$? For example, if ...