# 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.

33,263 questions
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### 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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### 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 ...
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### 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 $[ \cdot ]$ ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
### 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 ...