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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.

3
votes
3answers
47 views

How to solve exponential equations like $2^x+x=5$?

I tried the following: Let $y=5-x$. Then, $2^{5-y}=y \implies y \cdot 2^y=32$ Taking the log of both sides yields $$\log_2 y + y = \log_2 2 + 4$$ And that's where I'm stuck.
0
votes
4answers
56 views

How to proceed with this math question?

This may seem elementary but I can't seem to find the right steps to take. $$ 3^a =21^b ~~~\mbox{and}~~~~ 7^c = 21^b $$ Proof that $$ b= \frac{ac}{a+c} $$
-4
votes
2answers
51 views

How to split middle term= $x^2+ 2\sqrt{5}x + 3$ [on hold]

It is from an algebraic equation. How to split middle term $=x^2 + 2\sqrt{5}x + 3$?
10
votes
2answers
104 views

Rational Exponents: why is $\frac{1}2$ undefined but $\frac{2}4$ is not?

I've been reviewing rational exponents and have this question. Given thhat $(-5)^\frac{1}2$ is undefined because this equals $\sqrt{-5}$ which again is undefined. Then why is it possible to solve $(-...
1
vote
5answers
79 views

real solution of equation $(x^2+6x+7)^2+6(x^2+6x+7)+7=x$ is

Number of real solution of equation $(x^2+6x+7)^2+6(x^2+6x+7)+7=x$ is Plan Put $x^2+6x+7=f(x)$. Then i have $f(f(x))=x$ For $f(x)=x$ $x^2+5x+7=0$ no real value of $x$ For $f(x)=-x$ $x^2+8x+7=0$...
0
votes
2answers
66 views

Find the value of $\alpha^{\frac13}+\beta^{\frac13}$

If $f(x)=x^2-5x+8, f(\alpha)=0$ and $f(\beta)=0$ then find the value of $\alpha^{\frac13}+\beta^{\frac13}$ $$\alpha+\beta=5$$ $$\alpha \beta=8$$ $$\alpha^{\frac 1 3}=\frac 2 {\beta^{\frac 1 3}}$$ ...
3
votes
1answer
82 views

Next Term Of Strange Sequence

I tutored a 10th grader and I was asked this puzzle and I had spent nearly an hour with it and got “no where”. Any one can crack it? Please let me know. Thank you. Question: Find the $14$ th term of ...
0
votes
1answer
21 views

The Lagrange Interpolation formula – Spivak's Calculus Ch 3 Problem 7(b)

The problem: Now find a polynomial function $f$ of degree $n - 1$ such that $f(x_i) = a_i$, where $a, \ldots, a_n$ are given numbers. I found that this question had been asked before, but I did not ...
1
vote
2answers
49 views

Closed form solution for constant exponent in sum

I am trying to solve for $\alpha$ in the following equation: $$ 0.80 = \frac{1}{3} \left( X_1^\alpha + X_2^\alpha + X_3^\alpha \right)$$ Right now I just use Excel and solver to find a numerical ...
2
votes
4answers
34 views

Extracting integral solutions from a quartic equation

The equation \begin{equation*} y^{4} + 4y^{3} + 10y^{2} + 12y - 27 = 0 \end{equation*} has two integral roots. Without resorting to the quartic formula, how would one extract the roots from it?
0
votes
1answer
36 views

Refactoring Amortization Formula

I've been trying to figure out how this amortization equation can go from what's on the left to the right (ref. Wikipedia). Took the same idea and came up with a generalized equation, but not sure why ...
-1
votes
1answer
44 views

Log Functions, solving for x [on hold]

Does anyone know how I could solve the equation below for $x$, algebraically? $$6(e^{-0.5x}-e^{-0.02x})=5$$ Thanks in advance.
2
votes
2answers
35 views

Finding range of $a$ in exponential inequality

If $a4^{\tan x}+a4^{-\tan x}-2=0$ has a real solution, where $0\leq x\leq \pi,x\neq \frac{\pi}{2},$ then interval of $a$ is Thoughts on that problem: Via the arithmetic-geometric inequality (AM-GM),...
0
votes
1answer
27 views

How to solve 3 linear equations in three variables using cross multiplication method?

How to solve 3 linear equations in 3 variables using cross multiplication method? I have no problem in solving these equations using substituting. However, how do I solve these using cross ...
0
votes
1answer
31 views

Simple percentage problem driving me crazy

Ok, so lets say to board a cruise ship it would usually take $60$ to $90$ minutes. Now it takes only $10$ minutes. In percentages this is: $60-10 = \frac{50}{60} = 83.3\%$ reduction (ie. from $60$ ...
5
votes
0answers
36 views

Does there exist a function $f_{\Box,\Box}(\Box)$ making the formula $a + (b \oplus c) = (f_{b,c}(a)+b) \oplus (f_{c,b}(a)+c)$ true?

Let $a$ and $b$ denote the resistances of two resistors. If they're put in series, the total resistance is $a+b$. If they're put in parallel, the total resistance is $$a \oplus b := \frac{1}{\frac{1}{...
-1
votes
1answer
31 views

Graph a system of equations

How to graph this system of equations: Also will be helpful if anyone can explain how to write it in Mathematica 12. I mean, how to use the { and to make columns and rows as it is shown This does ...
0
votes
0answers
28 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 ...
0
votes
1answer
22 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!!" ...
4
votes
0answers
61 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: ...
0
votes
2answers
38 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 $a, b, c,$...
-1
votes
1answer
23 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.
1
vote
2answers
46 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}...
0
votes
2answers
48 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
votes
1answer
36 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 ...
0
votes
5answers
60 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
votes
4answers
97 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 ...
0
votes
3answers
40 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 ...
2
votes
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 ...
2
votes
7answers
68 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 $a$ ...
2
votes
4answers
66 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 $(x+\frac{b}{2a})$ without absolute value ...
0
votes
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
187 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
votes
2answers
42 views

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

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 ...
0
votes
2answers
24 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 - $1) \frac{n(n+1)(n^2+n+1)}{18}$ $2)\frac{n(n+1)(n^2+n+1)}{6(2n+1)}$ $3) \frac{n(n^2+n+1)}{3(2n+1)}$ $4) \frac{n^4+2n^3+2n^2+n}{3(2n+1)}$ $5) \frac{n(n+1)(2n^2+n+1)}{2(2n+1)}$
0
votes
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
votes
2answers
105 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
votes
7answers
104 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
votes
2answers
28 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 ...
-1
votes
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.
0
votes
1answer
56 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
votes
3answers
95 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^...
0
votes
1answer
13 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 $K[X]$....
1
vote
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+\...
0
votes
1answer
59 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
vote
1answer
40 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 ...
-1
votes
2answers
59 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
votes
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
vote
2answers
39 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}=\...
-2
votes
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.