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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.

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How can I try myself to solve exponential equations easily?

I spent hours trying to solve: $$4^x + 1 = 2^{x+1}$$ Can you guide me on how to solve this? How can I train myself to always find the right "trick" to solve such equations? Rather than just practicing... Is there a better way to always know how to…
Marcus Wayne
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Number of real solutions

Prove that the equation $\lfloor x\rfloor+\lfloor 2x\rfloor+\lfloor 4x\rfloor+\lfloor 8x\rfloor+\lfloor 16x\rfloor+\lfloor 32x\rfloor = 12345$ does not have any real solution. ($\lfloor x\rfloor$ denotes the greatest integer less than or equal to…
Maverick
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Solve equation: $25x+9\sqrt{9x^2-4}=\frac{2}{x}+\frac{18x}{x^2+1}$

Solve equation: $25x+9\sqrt{9x^2-4}=\dfrac{2}{x}+\dfrac{18x}{x^2+1}$ I used wolframalpha.com and got the only solution $x=-\dfrac{1}{\sqrt2}$ And this is my try: Domain: $|x|\ge\dfrac{2}{3}$ If $x\ge\dfrac{2}{3}$, we…
idiots
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Evaluating the infinite product $ \prod\limits_{n=1}^{\infty} \cos(\frac{y}{2^n}) $

How does one evaluate $ \displaystyle\prod\limits_{n=1}^{\infty} \cos(\frac{y}{2^n}) $? Seems fairly straightforward, as I just plugged in some initial values $n = 1, 2, 3,\dotsc$ $n = 1$ $ \sin(y)= 2\sin(\frac{y}{2})\cos(\frac{y}{2})$ $…
VladeKR
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A calculator is broken so that the only keys that still work are the basic trigonometric and inverse trigonometric functions

A calculator is broken so that the only keys that still work are the $\sin$, $\cos$, $\tan$, $\cot$, $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$ buttons. The display initially shows 0. (Assume that the calculator does real number calculations with…
SAM
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"Rationalizing" an equation

$$x=\sqrt[3]{p}+\sqrt[3]{q}$$ I'm trying to figure out some way to "rationalize" the previous equation, meaning to rewrite it purely in terms of whole number powers of $p$, $q$, and $x$. It seems quite simple but I've been stuck trying to do it. I'd…
quote
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Finding general solutions of trigonometric equations

I have to find general solutions for $$\cos(x) + \cos(2x) = 0.$$ From sum-to-product formula I got $$\cos(3x/2) \cos(x/2) = 0,$$ giving me $$x = \pi/3$$ and $$x = \pi.$$ According to my text book, the answer should be in the form $$2n\pi \pm…
Lax
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Is it possible to solve a (simple) problem that includes remainders with basic algebra?

An eight-year old (Grade 3) told me about the "hardest problem" they had to solve on their math test yesterday. Here's the question: There is a number less than 40, that when divided by 5 leaves a remainder of 3, and when divided by 6 leaves a…
Jedidja
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Sign function using only basic arithmetic: $+, -, \times , \div$

I wonder if it is possible to build a $\operatorname{sign} (x)$ function which will return either $1$ for positive values of $x$, or $-1$ for negative ones - using only the four basic arithmetic operations $(+, -, \times , \div )$. Clearly it is…
Brainless
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Number of positive solutions?

What is the number of positive solutions to $$ (x^{1000} + 1)(1 + x^2 + x^4 + \cdots + x^{998}) = 1000x^{999}? $$ I tried to solve it. First I used by using sum of Geometric Progression. Then the equation becomes too complicated and is in the…
vikiiii
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Rearranging a formula, transpose for A2 - I'm lost

Given the formula: $$ q = A_1\sqrt\frac{2gh}{(\frac{A_1}{A_2})^2-1} $$ Transpose for $A_2$ I have tried this problem four times and got a different answer every time, none of which are the answer provided in the book. I would very much appreciate if…
mal
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If $|x|\leq 1$, $|ax^2+bx+c|\leq 1$, find the max possible value of $|2ax+b|$

Given $a,b,c\in \mathbb{R}$ such that $|x|\leq 1$ and $|ax^2+bx+c|\leq 1$, find the maximum value of $$ |2ax+b| $$ My Attempt: Set $x=1$ in $|ax^2+bc+c|\leq 1$ to get $$ \tag{$\star$}|a+b+c|\leq 1 $$ Similarly, set $x=-1$ in $|ax^2+bc+c|\leq 1$…
juantheron
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Precalculus equation solving involving square roots

Solve the equation: $$\sqrt x = \frac{3}{\sqrt x}+ \sqrt {x+3}$$ My approach was to multiply both sides with $\sqrt x$: $$x = 3 + \sqrt {x+3} \sqrt x$$ $$(x - 3)^2 = (x+3)x$$ $$x^2 - 6x + 9 = 3x + x^2$$ $$9x-9 = 0$$ $$x = 1$$ ...but this is clearly…
MathInferno
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Find the sum of all odd numbers between two polynomials

I was asked this question by someone I tutor and was stumped. Find the sum of all odd numbers between $n^2 - 5n + 6$ and $n^2 + n$ for $n \ge 4.$ I wrote a few cases out and tried to find a pattern, but was unsuccessful. Call polynomial 1, $p(n) =…
mathamphetamines
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Prove the equation without solving for X

My niece asked me this - If $x=1/(5-x)$ prove $x^3 + \dfrac{1}{x^3}=110$ without solving for x. I said its not possible since solving for x itself gives me two roots for x (one being $\approx4.79$) and substituting for it in the second equation…
Vrashabh Irde
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