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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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1answer
45 views

Dividing Complex Numbers by Infinity

My PreCalculus teacher recently reviewed the properties of limits with us before our test and stated that any real number divided by infinity equals zero. This got me thinking and I asked them whether ...
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3answers
37 views

an algebraic question

If $a,b,c$ are nonzero and distinct real numbers, I have to determine if $$C=\frac {(a - c)^2}{c} + \frac{(a - b)^2}{a} + \frac{(b - c)^2}{b}$$ equals to zero. First, I can't apply any well-known ...
0
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0answers
12 views

Yield in terms of term

In return for an investment of an amount $P$ at time $0$, dividends of amount $c$ are received at times $1, 2, \ldots, n$, respectively, and a redemption payment of $R$ is received at time $n$. ...
0
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1answer
10 views

Stuck solving an ODE(IVP) with separation of variables

I am trying to solve an IVP which has a particularity of having several variables which should be found with a set of conditions, which is: $y'(t)=k y(t)(P-y(t))$ $y(0)=500$ $y(1)=1000$ $lim_{t-&...
2
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2answers
33 views

If $x^2+kx+1$ is a factor of $px^5+qx^2+r$ prove that $(p^2-r^2)(p^2-r^2+qr)=q^2p^2$

if $x^2+kx+1$ is a factor of $px^5+qx^2+r$ prove that $(p^2-r^2)(p^2-r^2+qr)=q^2p^2$ My try Since this is a factor I tired finding A, B, C & D such that, $(x^2+kx+1)(Ax^3+Bx^2+cx+D)$ =$px^5+qx^...
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1answer
59 views

How to split using partial fractions: $\frac{9x^2+48x+18}{(2x+1)(x^2+8x+3)}$?

I'm self-studying from Stroud & Booth's amazing "Engineering Mathematics", but am stuck on a problem from the "Partial Fractions" chapter. I've been running around in circles trying to solve it, ...
0
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2answers
46 views

I genuinely think WolframAlpha has it wrong in this case

Please look at the screenshot above. Why is $\displaystyle z(\pi/2)=\frac{1}{2}\sin \pi$ not showing as $0$?
1
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2answers
41 views

Definition of “ the principal n-th root of ” using a sign condition.[modified title]

[ Edited] Is it ok to say that, the principal n-th root of a is simply the number x such that : (1) x to the n-th power is equal to a and (2) x has the same sign as a ? My question deals ...
1
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2answers
67 views

If $ x(9^{\sqrt{x^{2}-3}} + 3^{\sqrt{x^{2}-3}}) = (3^{2\sqrt{x^{2}-3}+1} - 3^{\sqrt{x^{2}-3} + 1} - 18) \sqrt{x}+ 6x $, what is the maximum $x$?

If $$ x(9^{\sqrt{x^{2}-3}} + 3^{\sqrt{x^{2}-3}}) = (3^{2\sqrt{x^{2}-3}+1} - 3^{\sqrt{x^{2}-3} + 1} - 18) \sqrt{x}+ 6x $$ Whati is the maximum value $x$ that fits in the equation? Attempt: $$ \sqrt{x}...
0
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1answer
36 views

Anyone see how AM-QM inequality is supposed to be used here?

I'm going through a certain solution, which invokes using AM-QM inequality, but I don't see how. We are given a sequence $\{a_i\}$ of positive real numbers which for every $k$ satisfy $$ a_1+\ldots+...
11
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7answers
105 views

Solve $\sqrt{1 + \sqrt{1-x^{2}}}\left(\sqrt{(1+x)^{3}} + \sqrt{(1-x)^{3}} \right) = 2 + \sqrt{1-x^{2}} $

Solve $$\sqrt{1 + \sqrt{1-x^{2}}}\left(\sqrt{(1+x)^{3}} + \sqrt{(1-x)^{3}} \right) = 2 + \sqrt{1-x^{2}} $$ My attempt: Let $A = \sqrt{1+x}, B = \sqrt{1-x}$ and then by squaring the problematic ...
0
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3answers
39 views

Coefficient problem in algebra

Find the coefficient of $ x^{8} $ in the expansion of $ (1+x^2-x^3)^{9} $ I know the problem is simple if we use multinomial theorem and I got an answer $ 378 $ using it. Can someone check it and ...
1
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2answers
421 views

Is the chance of winning rock paper scissors $1/2$ or $1/3$? [on hold]

Is it $1/2$ or $1/3$. How do people think it is $1/2$? And what is the true answer, $1/2$ or $1/3$?
-4
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1answer
70 views

Discovered and proved 3 algorithms. Peer review request. Enthusiast-level. [on hold]

As some excellent choice of information has been recognized, I wrote a proof to create an algebraic equation which fits a form of tangent, sine, and cosine. It's very simple. I can explain the roles ...
1
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3answers
56 views

Prove: $\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x}=0$ has a real root between $\frac13a$ and $\frac23a$, and one between $-\frac23b$ and $-\frac13b$

If $a$ and $b$ are positive numbers, prove that the equation $$\frac{1}{x-a}+\frac{1}{x-b}+\frac{1}{x}=0$$ has two real roots; one between $\frac13a$ and $\frac23a$, and one between $-\frac23b$ ...
1
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3answers
47 views

How to find this kind of relationship: $x-\frac{1}{x}=A$ and $x+\frac{1}{x}=\sqrt{A^2+4}$?

Could someone explain how to get from: $x-\frac{1}{x}=A$ to $x+\frac{1}{x}=\sqrt{A^2+4}$ ? It is one of the Algebra II tricks. Thanks.
4
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5answers
74 views

Question about the proof of simple algebra rule $\frac{1}{\frac{1}{a}} = a$

I have a question about a proof I saw in a book about basic algeba rules. The rule to prove is: \begin{eqnarray*} \frac{1}{\frac{1}{a}} = a, \quad a \in \mathbb{R}_{\ne 0} \end{eqnarray*} And the ...
0
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0answers
46 views

Cant understand ram formula

Im trying to get how ram`s damage of a wall is calculated in travian. I saw this formula: https://wbb.forum.travian.com/index.php?thread/75248-combat-system-formulas/ under ram, it states: ...
0
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3answers
43 views

How is my book completing the square?

In my book I see: To integrate the given function we complete the square in the denominator: $$4x^2 - 4x + 3 = (2x-1)^2 + 2$$ How is it doing this? When I complete the square I get: $$x^2 - x + ...
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2answers
22 views

A machine produces B unit goods in A hours… [on hold]

A machine produces $b$ unit goods in $a$ hours. How many hours does this machine need to produce $b\cdot c$ unit goods?
1
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2answers
24 views

complete set of values of $a$ having modulus and linear terms

If $(9-x^2)>|x-a|$ has at least one negative real solution for $a\in\mathbb{R}.$ Then complete set of values of $a$ is Plan If $x>a$ Then $9-x^2>x-a\Rightarrow x^2+x-(a+9)<0$ If $x\...
1
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2answers
71 views

Find parameter a for which..

Find parameter $a$ for which $$\frac{ax^2+3x-4}{a+3x-4x^2}$$ takes all real values for $x \in \mathbb{R}$ I have equated the function to a real value, say, k which gets me a quadratic in x. I have ...
0
votes
0answers
20 views

Solve $W_0(x)^2 - W_{-1}(x)^2 = c$ for $x<0$ with sufficiently large $c>0$

Suppose we have $$ W_0(x)^2 - W_{-1}(x)^2 = c $$ for some constant $c>0$ with $x<0$. Then can we solve for $x$ algebraically? Or at least analytically find bounds for $x$? Here we can assume $...
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votes
2answers
23 views

Value of B in this ratio problem? [on hold]

a,b,c integers $a+b+c=45$ $\frac{a}{b}=\frac{3}{2}$ $\frac{b}{c}=\frac{4}{5}$ b=?
0
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1answer
39 views

Consider a function, . If the function is horizontally translated 3 units to the left and two units down, the new function would be expressed as

Consider a function, . If the function is horizontally translated 3 units to the left and two units down, the new function would be expressed as y=f(x+3)-2 is it ok Thank you
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1answer
50 views

How many roots are there for the polynomial given above? [on hold]

How many roots are there for the polynomial given above? Is it three or im wrong
0
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3answers
53 views

How to solve this inequation?

So the inequation is this $x^{2016}-1<0 $ My initial idea was to transform it like this $x^{2016}>x^0$ and then to look four cases: $1.$ when $x \lt 0\lt 1$ $2.$ when $x \gt 1$ $ 3.$ when $-...
0
votes
1answer
39 views

Find the solutions of the system of equations $a+b= c^n$;$b+c=a^n$;$c+a= b^n $,$\forall a,b,c \in \mathbb R$ and $n \in \mathbb Z^+$

Find the solutions of the system of equations $a+b= c^n$;$b+c=a^n$;$c+a= b^n $,$\forall a,b,c \in \mathbb R$ and $n \in \mathbb Z^+$ Let $a,b$ and $c \in \Bbb R$ and $n$ a nonnegative interger. Find ...
1
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1answer
62 views

Reasoning behind integrating f(x)/g(x)?

I understand the method to integrate this function would be: $\int{\frac{x^2+1}{x^4-x^2+1} \thinspace dx}$ Divide all terms by $x^2$: $= \int{\frac{\frac{x^2}{x^2}+\frac{1}{x^2}}{\frac{x^4}{x^2}-\...
0
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2answers
42 views

Radical equation and extraneous solution: -4 or 3 for $\sqrt{12-x}=x$

I have a radical equation to solve: $\sqrt{12-x}=x$ Before checking for extraneous solutions I arrived at -4 and 3. My textbook says the only solution is 3 but surely it's -4 too? $\sqrt{12-(-4)}=-...
0
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1answer
21 views

The power and the force of an engine

A car moves on an inclined plane with an angle whose sine is $\frac{1}{40}$ with uniform velocity 10 m/s against resistances 1225 N .Find the power of the engine .and if the power increased suddenly ...
-2
votes
2answers
42 views

Write $f(x)$ as a product of two polynomials [on hold]

Consider the polynomial $f(x)=8x^3-50x^2+17x-30$. If $f(6)=0$ write $f(x)$ as a product of two polynomials
0
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3answers
17 views

Finding the positive \ negative domain of a simple expression

How can I find the right domain of a simple expression, I tried to check the positive domain of this function: $$ \frac{{900-6X} }{X+50} > 0$$ $$900-6X > 0 $$ $$900 > 6X$$ $$ 150 > X$$...
0
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1answer
52 views

What functions MUST be written in the form $y = a f(b(x-h)) + k$ [on hold]

In my book I saw that the general transformations of all functions can be written in the form $y = a f(x-h) + k$. However I saw the most periodic functions must be written in the form $y = a f(b(x-h)...
0
votes
1answer
27 views

Evaluating the argument of a complex expression

I am trying to verify a relationship presented in this paper. I want to verify that the argument of $$\frac{E_{2}}{E_{1}}=\frac{r-\tau\exp\left(i\varphi\right)}{1-\tau r\exp\left(i\varphi\right)},...
0
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4answers
54 views

Equation whose solution is a finite tower of $2's$

What equation has a finite power tower of $2's,$ as it's solution: $$ x=2^{{2^{2}}^{\cdot\cdot\cdot}}.$$ I tried to reverse engineer the solution, back into an equation, so I started with a toy ...
1
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0answers
12 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 ...
1
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2answers
76 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
51 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
42 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
46 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
60 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 ...
0
votes
3answers
45 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, ...
0
votes
1answer
39 views

Problem on parabola from conics [on hold]

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 ...
0
votes
1answer
60 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 ...
0
votes
2answers
41 views

Quadrants of points located in the x and y axes [on hold]

Which quadrant are the points that lie on the axis in? (e.g. the points $(0, 2)$ or $(4, 0)$)
7
votes
2answers
130 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....
-2
votes
2answers
63 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 ...
0
votes
1answer
25 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 $...
1
vote
2answers
45 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 $ ...