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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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2 views

In-Depth on Function Transformations

If all functions can be written in the form y = af(x-h)+k, how does this account for horizontal stretching and compression? Why is the default form for all functions not y = af(b(x-h))+k. I know of ...
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2answers
26 views

How do I prove the inequality $(\sum a^3)^2 \leq (\sum a^2)^3$?

Let $a_1, \dots, a_n \in \mathbb{R}.$ I wish to show that $(\sum_{i=1}^n a_i^3)^2 \leq (\sum_{i=1}^n a_i^2)^3$ in order to prove another statement. But I cannot see how to prove this, if at all the ...
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2answers
32 views

Solving a polynomial by grouping and factoring - why does this answer have $\pm3i$?

I am asked to solve for x in the polynomial using factoring and grouping: $5X^3+45X=2X^2+18$ My working: $5X^3-2X^2+45X-18$ $X^2(5X-2)+9(5X-2)$ $(X^2+9)(5X-2)$ So: $X^2+9=0$ $X^2=-9$ $X=i\sqrt{...
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0answers
37 views

Solving exponential and logarithm mixed together

Exp(-x)=Cosx. So i shifted the exponent to the right hand side. 0=exp (x)Cosx. Got stuck here. Don't know whether to solve them individually. Like Cosx =0 Exp (x)=0
2
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4answers
43 views

Long division of $\frac{3x^3-x^2-13x-13}{x^2-x-6}$

I'm self-studying from Stroud & Booth's amazing textbook "Engineering Mathematics", and am on the "Partial Fractions" chapter. As part of an exercise I need to do long division of two polynomial ...
0
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1answer
51 views

Does this equation yield only primes?

Interested in solving this equation for $x$: $\exp\Big(\frac{n}{\ln(\pi(x))}\Big)=\pi(x)$ for $n=1,2,3,...$ For $n=1$ up to $n=9,$ I got $x=5,11,13,19,29,37,47,59,73.$ $\pi(x)$ is the prime ...
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3answers
39 views

How to evaluate $\sum_{ r=1}^{16}(5r-7)$?

I'm self-studying from Stroud & Booth's "Engineering Mathematics" and in the "Binomials" chapter, one of the last exercises is to evaluate: $$\sum_{ r=1}^{16}(5r-7)$$ This has got me confused, ...
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1answer
36 views

Problem on parabola from conics

An arch-shaped monument is often mistak- en to be parabolic in shape. In fact, it is a catenary, which has a more complicated formula than a parabola.The arch is 475 feet high and 444 feet wide at ...
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1answer
57 views

Which square should be cut to minimize loss?

From a paper size of $950mm × 1200 mm$, squares with a side of $64 mm$ or $46 mm$ can be cut. Which square should be cut to minimize loss? My attempts: We have, for square with side 64 mm, the ...
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2answers
36 views

Quadrants of points located in the x and y axes

Which quadrant are the points that lie on the axis in? e.g. the points (0, 2) or (4, 0).
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2answers
115 views

How can i Prove that the gray area is the same as white area? [duplicate]

A circle is cut into 8 parts, each part has the angle 45 degrees from an arbitrary point. how to prove that the white area is the same as the Gray area? I just want any hint for solving this question....
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2answers
55 views

Solve for x in $ x^2 + y^2 = 1 $ and $ x \pm y = \frac \pi4 $

Solve for x in $ x^2+ y^2 = 1 $ and $ x \pm y = \frac \pi4 $ I tried solving this by substitute method. And using the quadratic formula, but that create lots of cases. The original problem was to ...
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1answer
23 views

Inconsistent answers from inferring probability of success from probability of failure

Alright so I was working on my previous post and stumbled into a problem. Say the $P(A$) failing is $0.02$, which translates to $2\%$ failure rate. Say the P(B) failing is 0.003, which translates to $...
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2answers
39 views

prove that $ 2 \arctan({\csc \arctan x - \tan \text{arccot }x}) = \arctan x $

Prove that $ 2 \arctan({\csc (\arctan x) -\tan (\text{arccot }x)}) = \arctan x $ x is not equal to zero. So, to solve this I tried I made two condition $ x \gt 0 $ and $ x \lt 0 $ If $ x \gt 0 $ ...
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0answers
24 views

Rationalizing denominator with any number of radicals

I'm trying to develop a java class for exact algebraic numbers. I've come to a little bit of a roadblock as far as this goes. My question right now is how to rationalize these equations, but no-one I'...
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2answers
47 views

When M is negative one should prefer $\sqrt{|M|}$ to $\sqrt{-M}$. Right? [on hold]

For otherwise we get $\sqrt{-M}=\sqrt{-1}\cdot\sqrt{M}$ but the terms on the right are meaningless.
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3answers
63 views

Find the solution of y = f(x)

Find the solution $y = f(x)$ of: $$ x^2 + y^2 - x^3 = 0 $$ Near the following points $(5,10)$,$(10,-30)$ I think I need to use the implicit function theorem and I tried this First for the ...
4
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4answers
71 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.
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4answers
67 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 $$ b= \frac{ac}{a+c} $$
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2answers
54 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 $=x^2 + 2\sqrt{5}x + 3$?
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2answers
116 views

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

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 $(-...
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6answers
93 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 $f(x)=-x$ $x^2+8x+7=0$...
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2answers
69 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 $\alpha^{\frac13}+\beta^{\frac13}$ $$\alpha+\beta=5$$ $$\alpha \beta=8$$ $$\alpha^{\frac 1 3}=\frac 2 {\beta^{\frac 1 3}}$$ ...
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1answer
92 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 ...
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1answer
22 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
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3answers
69 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
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4answers
35 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?
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1answer
37 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 ...
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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
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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),...
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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 ...
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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 $60$ ...
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0answers
39 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}{...
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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 ...
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0answers
29 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
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!!" ...
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0answers
65 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
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 $a, b, c,$...
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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.
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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}...
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2answers
49 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
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 ...
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5answers
62 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
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
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3answers
41 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
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1answer
30 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
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7answers
69 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
72 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
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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
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5answers
196 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 ...