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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 $e^{i \pi} = -1$

Possible Duplicate: How to prove Euler's formula: $\exp(i t)=\cos(t)+i\sin(t)$ ? I recently heard that $e^{i \pi} = -1$. WolframAlpha confirmed this for me, however, I don't see how this works.
mbreedlove
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Proving $1\times 1! + 2\times 2! + 3\times 3! +\cdots.+n\times n! = (n+1)! -1 $

Prove that: $$1\times 1! + 2\times 2! + 3\times 3! +\cdots+n\times n! = (n+1)! -1 $$ My attempt :- Let S= $1\times 1! + 2\times 2! + 3\times 3! + ...+n\times n! = (n+1)! -1 $ = $1\times 1! + 4\times 1! + 9\times 2! + 16\times 3! + ...+ n^2…
Hussien Mohamed
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Mathematical Expressions; School Homework

$y + 3$ is always $5$ more than $y – 2$ so $y + 3 – (y – 2) = 5$ $(y + 4 ) – (y – 3) = ~ ?$ $(y - 2) – (y - 3) = ~ ?$ How would you work it out? I know that $y + 3$ is 5 more than $y - 2$, so should I do trial method or is there a method that's…
user61406
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Field of study dedicated to algebra 'tricks' such as change of variable, obscure identity substitution, etc.?

I'm an undergraduate math major and sometimes I find proofs that seem to use algebraic 'tricks' to reach their conclusions. The 'trick' that I see most often is a change of variable or the use of an obscure identity, and intuitively I have no idea…
Battlefrisk
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If $\frac{1}{a+3}+\frac{1}{b+4}+\frac{1}{c+5}=\frac{7}{12}$ for positive integer $a$, $b$, $c$, then find $\frac{a}{a+3}+\frac{b}{b+4}+\frac{c}{c+5}$

Given $a, b, c$ are positive integers, and that $$\frac{1}{a + 3} + \frac{1}{b+4} + \frac{1}{c+5} = \frac{7}{12},$$ compute: $$\frac{a}{a+3} + \frac{b}{b+4} + \frac{c}{c+5}$$ Source (Romanian Math Magazine, Gazeta Matematica S:E19.333, this…
Mircea
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Is there a way to prove that $x > \sin^2(x) , x >0$

The inequality of $$ x > \sin^2(x), \forall x \in ( 0 , + \infty) $$ can be visualized, and easily, we can prove that it's true (via the graph). However, how can we prove that it's true algebraically?
average_discrete_math_enjoyer
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Calculations with Percentages

A group has 60% female and 40% male members. 42% of all group members are older then 60 years, among men even 60% are older then 60. a) which percentage of the women is younger then 60? b) If a young group member (i.e. under 60 years) enters the…
TestGuest
  • 1,053
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$a,b$ and $c$ are roots of $x^3-x-1=0$. Find $a^{\frac{2}{3}}+b^{\frac{2}{3}}+c^{\frac{2}{3}}$

$a,b$ and $c$ are the roots of $$x^3-x-1=0$$ Find $$a^{\frac{2}{3}}+b^{\frac{2}{3}}+c^{\frac{2}{3}}$$ To solve this question I called the quantity A and I calculated $A^3$. I have a feeling there is a better way. Question From Jalil Hajimir
Kian
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How many natural numbers $n$ exist for $n=a^2−b^2−c^2$

How many natural numbers $n≤1000$ cannot be written in the form $a^2−b^2−c^2 \ ;$ where $a$,$b$ and $c$ are non-negative integers subject to condition $a≥b+c$. How to approach?
ABC
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Roots of equation form infinite sequence

The sequence $a_n$ has the property that $a_n$ and $a_{n+1}$ are the roots of the equation $$x^2-c_nx+\frac{1}{3^n}=0$$ and $a_1=2$. What is $\sum_{n=1}^{\infty}c_n?$ By Vieta's, $a_{n+1}=\frac{1}{3^na_n}$ and $c_n=a_n+a_{n+1}$. Additionally…
Baker013273213
  • 862
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Deducing information based on common denominators between unknown fractions.

I'm trying to answer a 3-part question from the "Art of Problem Sovling: Introduction to Algebra" book. The problem is outlined as: Fiona, George and Henry each think of a different fraction. The simplest common denominator between Fiona's and…
Vocaloidas
  • 283
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If $(\pm x)^2 = X$ then $\sqrt{X} = x$?

I've been helping my siblings with their GCSE and A Level maths and I've come across a question where they have just taken the positive square root. It's a pure maths question and there's no (obvious) reason to ignore the negative square root. I…
Kaish
  • 6,126
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Calculate the number of solutions

Calculate the number of solutions presented by the equation. $$\sqrt{1-x}+\sqrt{1-2x}+\sqrt{1-4x}=x^2+2$$ What I thought: The LHS is concave, the RHS is convex
Lambert macuse
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If volume of a sphere increases by 72.8%, what is change of its surface area?

If the volume of a sphere is increased by 72.8% what would be the change in surface area? I'm trying to solve the problem using application of derivatives. I noticed that on differentiating the formula for volume of a sphere we directly end up with…
Techie5879
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Solve in the positive real number the system of equation.

Solve in the positive real numbers, the system of equations : $$(2x)^{2013} + (2y)^{2013} + (2z)^{2013} =3 $$ and $$xy + yz + zx + 2xyz = 1.$$
Kushashwa Ravi Shrimali
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