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Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations and other symbolic-manipulation topics.

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0answers
8 views

Let y be a polynomial in p^1/n with rational coefficients.prove that in all polynomials y,y^2,y^3…y^n ,each term occurs in atleast 2 polynomials.

The above problem is part of a broader question that any polynomial with rational coefficients in p^1/n is the root of an equation of degree n with rational coefficients.proving the above leads to a ...
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1answer
16 views

How to deal with proportionality?

I have a big question: "If we know that $A = BC$ (so remark that $A$ is proportionnal to $B$) and $A = DE$, then $A^2 = BCDE$ or $A = \sqrt {BCDE}$. But then $A$ is not proportional to $B$ anymore!!" ...
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0answers
30 views

$a$ and $b$ are solutions of $ \frac{1}{x^{2} - 10x-29} + \frac{1}{x^{2} - 10x-45} - \frac{2}{x^{2} - 10x-69} = 0 $, $a+b=?$

$a$ and $b$ are solutions of $$ \frac{1}{x^{2} - 10x-29} + \frac{1}{x^{2} - 10x-45} - \frac{2}{x^{2} - 10x-69} = 0 $$ What is $a+b=?$ $$ $$ Are there better approaches than the one below? Solution: ...
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2answers
30 views

If $a, b, c,$ and $d$ are integers and $\frac{1}{a} + \frac{1}{b} = \frac{1}{c}+\frac{1}{d}$ where both sides of the equation be the same

If we have integers $a$, $b$, $c$, $d$ and $\frac{1}{a} + \frac{1}{b} = \frac{1}{c}+\frac{1}{d}$,when will both sides of the equation be the equal. So what that means is that for integers $a, b, c,$...
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1answer
22 views

How to prove $t_k\geq \frac{k+1}{2}$ [on hold]

Let $t_1 = 1$, $t^2_{k+1}-t_{k+1} = t^2_k$ for $k=1,2,...$, $t_k>0$. Prove that $t_k\geq \frac{k+1}{2}$. I have no idea of it, hope someone can cope with it. Thanks sincerely.
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2answers
40 views

Simplifying $\frac{1}{x^4+4x^2}$

I'm trying to solve this problem on my own and it involves simplifying the expression in the title. In the solutions it says it's this: $$\frac{1}{x^4+4x^2} = \frac{1}{4}\Biggl[\frac{1}{x^2}-\frac{1}...
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2answers
41 views

What's the value of $c$ that makes $|x - 3| < 1 \Rightarrow |x + 5| < c$ true?

In my Calculus lecture, we were given the following exercise to solve in class: Find the value of $c$ that makes $|x - 3| < 1 \Rightarrow |x + 5| < c$ true. This is what I came up with: $$\...
2
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1answer
32 views

How does this entropy equation simplify?

This is from the bok "Pattern Recognition and Machine Learning" By Bishop. I am having a hard time following the last step of this equation Where Stirlings approxaimation is subtituted for $\ln N!$ to ...
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5answers
54 views

Evaluating $\lim_{x\to 0}\frac{\ln(1+x)-\ln(1-x)}{\arctan(1+x)-\arctan(1-x)}$ without differentiation

$$\lim_{x\to 0}\frac{\ln(1+x)-\ln(1-x)}{\arctan(1+x)-\arctan(1-x)}$$ So, I have this limit and I'm trying to solve this limit without differentiation. I tried some steps, but they didn't come out ...
3
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4answers
89 views

Can someone show me how to solve $2^x + x = 4$?

I tried every way imaginable to solve the function $2^x + x = 4$, but I can't figure it out. I know it's not an easy one. I spend my free time helping people out with their math questions online, and ...
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3answers
38 views

How does $(x-1)^2 + 4 \leq 0$ tell us that $x^2 - 2x + 5 \geq 4$?

This is Example 3 on page 12 of "Calculus: One and Several Variables" by Salas, Hille, Etgen (10th edition). Solve the inequality $$x^2 -2x + 5 \leq 0 $$ The example proceeds to complete the ...
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1answer
29 views

If a, b belong to S prove that ab belongs to S

The question states that there is a set of real numbers S such that 1 belongs to S and if a, b belong to S, then a-b belongs to S and 1/a, 1/b will both belong to S. Prove that ab belongs to S. I ...
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7answers
51 views

Showing $a^2 + b^2 > 2ab$ without using the fact that $(a-b)^2 = a^2 + b^2 -2ab$?

I am wondering if we can Show that $a^2 + b^2 > 2ab$ without using the fact that $(a-b)^2 = a^2 + b^2 -2ab$? (I'm particularly interested in $0<a<b<1$ but I don't think restricting $a$ ...
1
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2answers
39 views

Is this valid when deriving quadratic equation?

When deriving the quadratic formula, isn't the square root of $(x+\frac{b}{2a})^2$ the absolute value of $(x+\frac{b}{2a})$? It's usually just represented as (x+(b/2a)) without absolute value and then ...
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2answers
24 views

Can you find an equation parallel or perpendicular to a line when it is not in slope intercept form?

For example, if I need to find the equation of the line parallel to $$2x-3y=4$$ which passes through the point $(1,-5)$ I know how to do this by putting it into slope-intercept form first to find the ...
7
votes
5answers
140 views

If $\sin(18^\circ)=\frac{a + \sqrt{b}}{c}$, then what is $a+b+c$? [duplicate]

If $\sin(18)=\frac{a + \sqrt{b}}{c}$ in the simplest form, then what is $a+b+c$? $$ $$ Attempt: $\sin(18)$ in a right triangle with sides $x$ (in front of corner with angle $18$ degrees), $y$, and ...
3
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2answers
39 views

Limit of $\sum_{r=0}^{n}\frac{{n\choose r}}{n^r\cdot(r+3)}$ as $n\rightarrow\infty$

Evaluate $$\lim_{n\rightarrow\infty}\sum_{r=0}^{n}\dfrac{{n\choose r}}{n^r\cdot(r+3)}$$ This form forcing me to use integrals, I tried expanding $${n\choose r}=\dfrac{n(n-1)\cdots (n-r+1)}{r!}$$ then ...
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2answers
22 views

If $n^{th}$ term of a series $T_n=\frac{n^4}{(2n-1)(2n+1)}$, find the sum of first n terms. [on hold]

The options are - $1) \frac{n(n+1)(n^2+n+1)}{18}$ $2)\frac{n(n+1)(n^2+n+1)}{6(2n+1)}$ $3) \frac{n(n^2+n+1)}{3(2n+1)}$ $4) \frac{n^4+2n^3+2n^2+n}{3(2n+1)}$ $5) \frac{n(n+1)(2n^2+n+1)}{2(2n+1)}$
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1answer
25 views

Solution set for $ 0 < | x - c | < \delta$. Is $|x - c| < \delta$ equivalent to $0 \leq | x - c | < \delta$?

Let $x \in \mathbb{R}$. Let $c$ be a real number constant, and $\delta > 0$ and also a real number. Consider the following: $$ |x - c| < \delta \label{1}\tag{1}$$ $$0 \leq | x - c | < \...
13
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2answers
89 views

If $x = \frac{\sqrt{111}-1}{2}$, calculate $(2x^{5} + 2x^{4} - 53x^{3} - 57x + 54)^{2004}$.

I already have two solutions for this problem, it is for high school students with an advanced level. I would like to know if there are better or more creative approaches on the problem. Here are my ...
7
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7answers
91 views

How to solve $\sqrt{x+2}\geq x$?

How do you solve the inequality $$\sqrt{x+2}\geq{x}?$$ Now since ${x+2}$ is under the radical sign, it must be greater than or equal to ${0}$ to be defined. So, ${x+2}\geq{0}$ Thus ${x}\geq{-2}$ ...
3
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2answers
27 views

splitting accommodation costs between people when some of them stay for fewer days

We're 4 people and we are staying 7 nights, for a total cost of 546. One of us however is leaving 1 day earlier. Initially I thought the problem was very simple. I reasoned that the 3 of us staying ...
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0answers
29 views

Can expression $y=\sqrt{x-x^2}$ be simplified? [on hold]

How can we simplify $y=\sqrt{x-x^2}$, in terms of $x$ ? Please describe the procedure.
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1answer
50 views

How to calculate area of triangle with perpendicular lines

In the figure below, the lines have slopes of 3 and 5. The lines intersect at (10,15). How far is it between the x-intercepts of the lines? The equation of the line with slope of 3 is $y=3x+b$ and ...
3
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3answers
91 views

How can I order these numbers without a calculator?

Classify the following numbers as rational or irrational. Then place them in order on a number line: $$\pi^2, -\pi^3, 10, 31/13, \sqrt{13}, 2018/2019, -17, 41000$$ I know $\pi$ is irrational so $\pi^...
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1answer
12 views

valuation of a composition of polynomials

Let $K$ be a field, $P$ be a irreducible polynomial of $K[X]$, $v_P$ be the valuation of $K[X]$ associated to $P$. Does one have $v_P(Q\circ R)=v_p(Q)\times v_P(R)$ for two polynomials $Q,R$ of $K[X]$....
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4answers
74 views

$x_1 x_2 x_3 x_4 + x_2 x_3 x_4 x_5 +…+ x_n x_1 x_2 x_3 = 0$ then what is $n$?

Can anyone please help me to understand what is the following problem saying?[! Each of the numbers $x_1,x_2,\cdots,x_n,n>4$, is equal to $1$ or $-1$. Suppose $$x_1x_2x_3x_4+x_2x_3x_4x_5+\...
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1answer
40 views

Showing that two numbers are the same percent different from their average.

More specifically, consider two real numbers $a,b>0$, and their average $r=\frac{a+b}{2}$. It is the case that $a=r*x$ and $b=r*y$ where $\vert 1-x\vert =\vert 1-y\vert$. For example, let $a=5$ ...
1
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1answer
38 views

Solving for $t$ in $h = 48t + \frac{1}{2}at^2$

I'm using an intermediate algebra textbook and it had this problem: "Solve the formula $$h = 48t + \frac{1}{2}a t^2$$ for $t$." The answer they displayed was: $$a = \frac{2h-96t}{t^2}$$ Can ...
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2answers
58 views

Upvotes and downvotes problematic. [on hold]

I just questioned myself this: Given that I have on a reddit post: $5$ net upvotes $65\%$ upvote ratio. How many upvotes and downvotes do I have in total?. I think its solvable, but after some ...
0
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1answer
16 views

Division method for base-conversion

There is an article called “The Base-Conversion method:Why does it work?” (The Base-Conversion method:Why does it work?),which states that: Each time you divide, you're asking "Does the original ...
1
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2answers
37 views

Find the ratio $p:q:r$, if $p,q,r$ are in H.P, and their squares are in A.P.

If three unequal numbers $p,q,r$ are in H.P and their squares are in A.P, then find the ratio $p:q:r$ . Attempt A.P(1): $\dfrac{1}{p},\dfrac{1}{q},\dfrac{1}{r}$ $$ \dfrac{1}{q}-\dfrac{1}{p}=\...
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2answers
21 views

Indices Question & Equation [on hold]

Following is the equation $$x^{x\sqrt x}=x\sqrt{x}$$ We need to find $x$. Please help.
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0answers
67 views

Eliminate $\alpha,\beta,\gamma$ from the system of equations

Eliminate $\alpha,\beta,\gamma$ from the following system of equations. $$a\cos(\alpha)+b\cos(\beta)+c\cos(\gamma)=0$$ $$a\sin(\alpha)+b\sin(\beta)+c\sin(\gamma)=0$$ $$a\sec(\alpha)+b\sec(\beta)+c\...
0
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1answer
32 views

Deriving simple formula for summation

I am first year non-math student and I am trying to formalize my homework from digital circuits course using some basic math tools. To give some initial context: If I have $\frac{1}{5}$ frequency ...
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2answers
33 views

How to calculate the complex fractional values with sum of numbers and letters in an equation

Below is the equation. Not sure if I am doing the right thing but i can't seems to get the right value. $$ \mathbf{1.6\over 121.8} = \frac { \mathbf{15\over M}} {\frac{15} M + {250\over 78} } $$ ...
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1answer
40 views

If $\lim_{t \rightarrow 2} \frac{ \sqrt[3]{a+ \frac{b}{t^3}-2 } }{t-2}=A$, then what is

If $\lim_{t \rightarrow 2} \frac{ \sqrt[3]{a+ \frac{b}{t^3}-2} }{t-2}=A$, then what is $\lim_{t \rightarrow 2} \frac{ \sqrt[3]{ \frac{a}{8} + \frac{b}{8t^3}}-t+1 }{t^2+2t-8}$ in terms of A? I ...
0
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2answers
20 views

Interrelated constraints via linear combinations

Given $x$ and $y$ are real variable such that: $\left| x \right|\le \alpha ,\left| y \right|\le \alpha ,$ where $\alpha$ is a positive constant. I want to determine bounds of $u,v$ where $u,v$ are ...
2
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1answer
70 views

Name for this method of factoring quadratic and are there any textbooks that describe it?

I remember learning this method of factoring quadratics in middle school or high school, but looking for a name or more information on it leads me to dead ends. Given: $ax^2+bx+c=0$ $d*e=a*c$ $d+...
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1answer
15 views

How would I set up an equation to find the 3 separate amounts of money invested?

Scenario: A total of $18,000 was divided and separated and to be invested in 3 different mutual funds, A, B, and C. Fund A promised to return 4% annual interest. Fund B, however lost 8% in annual ...
0
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1answer
26 views

Division algorithm for polynomials discrete maths

Problem: State the division algorithm for polynomials. Using this result, show that, if the polynomial $f(x)$ has a root $a$, then the linear polynomial $x-a$ divides $f(x)$. I’m incredibly stuck on ...
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votes
1answer
45 views

How is e^r shown from this? [on hold]

In this article, the author attempts to explain the effect of changing the value of r from 1 on the limit of $(1 + r/n)^n$ (how it changes from $e$) He approximates $e$ by $(1 + \frac{1}{100})^{100}$ ...
0
votes
2answers
38 views

Find for which lambda values this equation gives positive solutions [on hold]

The equation is $x^3 +x^2 +3 = \lambda x$ I thought I could move lambda to the first sector, getting $x^3 +x^2 -\lambda x+3 = 0$ But then? Can you show me the solution?
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1answer
37 views

Finding area using reimann sum [on hold]

$$ f(x)=ax-5x+10-a, \quad (1,2) $$ What would be limit of riemann sum? Can anyone give me answer with steps
0
votes
1answer
30 views

Trouble interpreting information to develop mathematical functions

Context: University coding assignment So far from this information, I've managed to develop two models: For the surface zone, I have $T(l)=\frac{-11}{45}l+24$ and for the deep zone, I have $T(l)=2$ ...
0
votes
2answers
48 views

need help with homework to find the missing number [on hold]

I was given this question for homework. I tried to resolve it by adding, subtracting the adjacent numbers, but could not figure out the answer. Below is the image of the problem: Any help will be ...
1
vote
6answers
51 views

Name of the arithmetic property if a=b then a+c = b+c?

Properties of arithmetic operations such multiplication and division have names. For example: $a + b = b + a$ (commutativity) $(a + b) + c = a + (b + c)$ associativity and so on is there a name ...
0
votes
1answer
33 views

How to think about/what is the justification for $\pm \sqrt{x^2} = x$?

It seems really strange that $\pm$ can be eliminated, as an algebraic manipulation... it doesn't seem like an algebraic rule. I think it's as simple as, if $x\ge0$, then $x = \sqrt{x^2}$, and if $x\...
0
votes
1answer
15 views

What does $0 < x_1 < x_2 < 1$ really mean when $y_1 = x_1/x_2$ and $y_2 = x_2$?

I am really having trouble seeing why $$0 < x_1 < x_2 < 1$$ is equaivalent to $$0 < y_1 < 1, \space 0<y_2<1$$ when $$y_1 = x_1/x_2, \space y_2 = x_2$$. The part that I am ...
0
votes
2answers
46 views

Find $x+y+z$ given that $x^{y^z} \cdot y^{z^x} \cdot z^{x^y} = 3xyz$

Find the value of $x+y+z$ (Given that $x,y,z \in \mathbb Z^+$) if $$x^{y^z} \cdot y^{z^x} \cdot z^{x^y} = 3xyz.$$ Any hint or idea will be very helpful, I tried my best but didn't get any approach.