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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.

0
votes
2answers
19 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{...
-1
votes
0answers
32 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
votes
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?
2
votes
4answers
42 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
votes
1answer
50 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 ...
6
votes
2answers
114 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....
0
votes
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).
0
votes
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 ...
0
votes
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, ...
1
vote
6answers
92 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
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 ...
0
votes
1answer
32 views

What does this dead end mean in a system of equations?

I have: $A = f_1(B,C,E) \quad \quad (1)$ $B = f_2(A,C,E) \quad \quad (2)$ $C = f_3(A,B,E) \quad \quad (3)$ $D = f_4(B,C,E) \quad \quad (4)$ $E = f_5(A,C,D) \quad \quad (5)$ where the $f()$s are ...
1
vote
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 ...
4
votes
4answers
69 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.
-2
votes
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 ...
0
votes
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 $...
0
votes
2answers
38 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 $ ...
6
votes
3answers
459 views

Looking for a website that teaches Geometry and Algebra II

I'm looking for a website that teaches Geometry and Algebra II greatly and emphatically. I'm beginning 9th grade in about a month and am trying to get a head start. I'm an extremely quick learner and ...
0
votes
1answer
40 views

Algebra of quadratic equation [on hold]

If $2$ lies between the roots of the equation $t^2 - mt + 2=0$, ($m$ belongs to $\Bbb R$) then what is the value of $$ \left[\bigg(\frac{3 |x| }{ 9 + x^2 }\bigg)^{\!m\,}\right], $$ where $[ \cdot ]$ ...
0
votes
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.
0
votes
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'...
0
votes
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 ...
13
votes
3answers
2k views

Division by imaginary number

I ran into a problem dividing by imaginary numbers recently. I was trying to simplify: $2 \over i$ I came up with two methods, which produced different results: Method 1: ${2 \over i} = {2i \over i^...
0
votes
3answers
154 views

Evaluate $\int_{0}^{\frac{\pi}{2}} \frac{\sin^2 nx}{\sin^2 x}\text{d}x$ [duplicate]

Evaluate $$ \int_{0}^{\frac{\pi}{2}} \frac{\sin^2 nx}{\sin^2 x} \text{d}x$$ where $n\in\mathbb{N}$ This one is another intriguing question from my worksheet. I'm only allowed to use ...
1
vote
1answer
113 views

Coefficients of rational involution.

Question: (Spivak Calulus 3rd, Chapter 3, Problem 8) For which numbers $a,b,c,d$ will the function $$f(x) = \frac{ax + d}{cx + b}$$ satisfy $f(f(x)) = x$ for all $x$? Attempt at an answer: I ...
0
votes
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 $$ b= \frac{ac}{a+c} $$
-4
votes
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 $=x^2 + 2\sqrt{5}x + 3$?
11
votes
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 $(-...
0
votes
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 ...
3
votes
1answer
91 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
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 $\alpha^{\frac13}+\beta^{\frac13}$ $$\alpha+\beta=5$$ $$\alpha \beta=8$$ $$\alpha^{\frac 1 3}=\frac 2 {\beta^{\frac 1 3}}$$ ...
1
vote
4answers
2k views

Probability :Knock Out Tournament Of Ranked Players

Thirty-two players ranked 1 to 32 are playing in a knockout tournament. Assume that in every match between any two players, the better-ranked player wins, the probability that ranked 1 and ranked ...
0
votes
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 ...
13
votes
1answer
332 views

Positive integer solutions of $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+\cdots+\frac{n}{a_n}=\frac{a_1+a_2+a_3+\cdots+a_n}{2}$

Find all ordered tuples of positive integers $(a_1,a_2,a_3,\ldots,a_n)$ such that $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+\cdots+\frac{n}{a_n}=\frac{a_1+a_2+a_3+\cdots+a_n}{2}$ The only thing I ...
3
votes
5answers
119 views

How to factorized this 4th degree polynomial?

I need your help to this polynomial's factorization. Factorize this polynomials which doesn't have roots in Q. $ \ f(x) = x^4 +2x^3-8x^2-6x-1 $ P.S.) Are there any generalized method finidng 4th ...
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 ...
7
votes
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 ...
1
vote
4answers
91 views

Solve the equation $\sqrt{x + 2} - \sqrt{3 - x} = x^2 - 6x + 9$.

Solve the equation: $$\sqrt{x + 2} - \sqrt{3 - x} = x^2 - 6x + 9$$ Here's what I've done. Let $\sqrt{x + 2} = a$ and $\sqrt{3 - x} = b$ $\implies \left\{ \begin{align} a^2 + b^2 &= 5\\ a^2 - b^...
-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
4answers
6k views

Whats the difference between Polynomial and Multinomial in two or more variables?

Whats the difference between Polynomial and Multinomial in two or more variables? Since, by definition: Multinomial: An algebraic expression having two or more (unlike) terms is called a ...
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 ...
7
votes
7answers
107 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 ${x}\geq{-2}$ ...
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 $60$ ...
0
votes
5answers
96 views

Solving $3\sqrt{7x-5}-4=8$

$$3\sqrt{7x-5}-4=8$$ On my homework, it said, "Solve each of the following radical equations algebraically. State any restrictions on the variable." I already solved the equation algebraically and ...
5
votes
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}{...
-2
votes
1answer
43 views

Proving an inequality with square roots

If got the following inequality which may or may not be true. But I think it should be true. I tried some inequalitys to get the roots away but I never got something which helped me: $3d+\sqrt{x_1^2+...
-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 ...
6
votes
0answers
96 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\...
2
votes
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 $a$ ...