Mathematics Stack Exchange
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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.

This tag is for questions typically taught in precalculus, as well as elementary algebra.

These topics include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomials, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

47234 questions
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Prove that $\frac{1}{a(a-b)(a-c)} +\frac{1}{b(b-c)(b-a)} +\frac{1}{c(c-a)(c-b)} =\frac{1}{abc}$ for all sets of distinct nonzero numbers $a,b,c$.

Prove that $$\cfrac{1}{a(a-b)(a-c)} +\cfrac{1}{b(b-c)(b-a)} +\cfrac{1}{c(c-a)(c-b)} =\cfrac{1}{abc}$$ for all sets of distinct nonzero numbers $a,b,c $. Now my question is not about how to solve this but rather why the technique which shows my…
Mr. Y
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sum of the Series $\sum^{n}_{r=0}(-1)^r\binom{n}{r}\left[\frac{1}{2^r}+\frac{3^r}{2^{2r}}+\frac{7^r}{2^{3r}}+........\bf{m\; terms}\right]$

The sum of the Series $\displaystyle \sum^{n}_{r=0}(-1)^r\binom{n}{r}\left[\frac{1}{2^r}+\frac{3^r}{2^{2r}}+\frac{7^r}{2^{3r}}+........\bf{m\; terms}\right]$ $\bf{My\; Try::}$Let $$\displaystyle S=…
juantheron
  • 53,015
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Inverse of $4-\ln(x+2)$

How to find an inverse of function $4-\ln(x+2)$? I know that inverse of $\ln(x)$ equals $e^{x}$, then inverse of $-\ln(x)$ equals $e^{-x}$ but what to do with the other numbers (4 and 2)?
aagaa
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Minimize $\;\left(-x+y+1 \right)^2 + \left( x-y-2\right)^2 + \left(x+2y-3 \right)^2 \;$ without using partial derivatives

How to find minimum of the expression $$\, \big(\!-x+y+1 \big)^2 + \big( x-y-2\big)^2 + \big(x+2y-3 \big)^2 \,$$ without using partial derivatives? It is easy to find the answer $\; x = 2, \; y = \dfrac{1}{2}\; $ by computing gradient of the…
Vlad
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Explain this logic to me please

$$ \ \frac{3}{(x^2+4)(x^2+9)} = \frac{Ax + B}{(x^2+4)} + \frac{Cx+D}{(x^2+9)} $$ Instructions say that "we can anticipate that $$ A = C = 0,$$ because neither the numerator nor the denominator involves odd powers of x, whereas nonzero values of A…
ababzy
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Prove that the coefficients of a quadratic function with real roots cannot be in geometric progression

Suppose $$ax^2+bx+c$$ is a quadratic polynomial (where $a$, $b$ and $c$ are not equal to zero) that has real roots. Prove that $a$, $b$, and $c$ cannot be consecutive terms in a geometric sequence. I tried writing the geometric sequence as $$a,\…
Jonathan
  • 514
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geometric mean of negative numbers is positive or negative?

What is the geometric mean of $-1$ and $-16$? Should it give $-4$ or $+4$?I think it should be $-4$ because a mean should always be greater than the least number and less than the greatest number. But square root always gives positive value, so what…
Pritam Ghoshal
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Why does fixed point iteration only produce the solution greater than $1$ to the equation $Mx = e^x$ for $x \in \Bbb R$?

The equation $Mx = e^x$, when $M > 0$. I know that the first solution must be at the tangent where the line $Mx$ crosses $e^x$, so $M$x has gradient $e^x$. This leads to $x(e^x) = e^x$, $x = 1$ But all of the $M$ values greater than $e$ must yield…
jg mr chapb
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Is there a rule for $\sqrt{a+b}$?

You learn in algebra that $$\sqrt{ab}=\sqrt{a} \sqrt{b}$$ and that $$\sqrt{\frac ab}=\frac {\sqrt a}{\sqrt b}$$ You also learn to never make the fatal mistake of thinking $$\sqrt{a+b}=\sqrt{a}+\sqrt{b}$$ However, I am wondering if there is a rule…
Tdonut
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what is an easy way to find the asymptotes of the graph $r = \frac1{1+2\cos \theta}?$

i would like to know an easy way to find the asymptotes of the $$r = \frac1{1+2\cos \theta}.$$ this is for a precalculus audience, so it will be nice if calculus can be avoided altogether. i know that the asymptotes are parallel to the lines…
abel
  • 29,170
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How I can find the value of $abc$ using the given equations?

If I have been given the value of $$\begin{align*} a+b+c&= 1\\ a^2+b^2+c^2&=9\\ a^3+b^3+c^3 &= 1 \end{align*}$$ Using this I can get the value of $$ab+bc+ca$$ How i can find the value of $abc$ using the given equations? I just need a hint. I…
vikiiii
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Rewriting $\prod_{i=1}^{n}\frac{x_i}{(1+x_i)^2}$ as $\frac{\prod_{i=1}^{n}x_i}{\prod_{i=1}^{n}(1+x_i)^2}$

Can we rewrite $$\prod_{i=1}^{n}\frac{x_i}{(1+x_i)^2}$$ as $$\frac{\prod_{i=1}^{n}x_i}{\prod_{i=1}^{n}(1+x_i)^2}$$ ?
ABC
  • 1,397
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Blending values on the number line

If we consider a portion of the number line, say on the interval $[0,100]$, and split that into regions e.g. split at $80$ to create $2$ regions. Now I want to subdivide the two regions. The region $[0,80]$ has $m$ partitions and the region…
PeteUK
  • 1,570
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How long does it take the average voter to vote?

So I was helping my brother with his homework question as follows The voting office can handle $50 \space \text {voters/hour}$ and has 20 voting stations. How long does it take the average voter to vote? He answered saying there are $60$ minutes for…
Jeff
  • 123
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3 answers

Solving the equation $- y^2 - x^2 - xy = 0$

Ok, this is really easy and it's getting me crazy because I forgot nearly everything I knew about maths! I've been trying to solve this equation and I can't seem to find a way out. I need to find out when the following equation is…
Schiavini
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