# Tagged Questions

Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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### For what values of $a,b,c$ will $ax^2+bx+c \geq 0$ hold $\forall x \in \mathbb{R}$?

If I let $y=ax^2+bx+c, (a\neq 0)$ then extremum of $y$ is attained at $x=-\frac{b}{2a}$. Then $\large\frac{\mathrm {d^2}y}{\mathrm {d}x^2}\big|_{(x=-\frac{b}{2a})}=2a$ which is positive or negative ...
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### Please help me solve this $x(\sqrt{2x+5}+\sqrt[3]{7x+13}) = 3x+6$

Wolfram Alpha shows that the answer is $x=2\,$ and $x=-2\,$ but what would be the best way of simplifying this equation ? It has been many years since I was in school , and I just cannot wrap my head ...
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### prove of sum of n real numbers greater than n given product is one [duplicate]

If the product of $n$ positive real numbers is $1$.Then prove that their sum is never less than $n$.
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### What is the formula called that calculates the scales of things?

I use this formula all the time: $$\frac{10}{100} = \frac{100}{x}$$ For example, the question may be, "if 10 apples equal 100 slices than 100 apples equals how many slices?" Is there a name for ...
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### Changing forms with factoring expressions [closed]

I need to know how to change between the quadratic forms: $ax^2+bx+c$ and $m(x+a)(x+b)$ by factoring trinomials given the values of $a, b$ and $c$ in the first form ($a$ and $b$ are not equivalent in ...
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### How to solve for log with a number outside?

$$\log_6(4x-10)+1 = \log_6(15x+15)$$ This is a sample problem. I know that when the bases of log are the same, all you have to do is set the parenthesis inside equal to each other. If the $1$ wasn't ...
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### Solving irrational inequality

Given inequality $(x - 2)\sqrt{x^2 + 1} > x^2 + 2$, find it's solution as intervals. And I have problem solving it. So at first, both $\sqrt{x^2 + 1} > 0$ and $x^2 + 2 > 0$. That means, that ...
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### expansion of a generating function

I found this formula in a book $$\sqrt{1-4x}=1-2\sum_{n=1}^{\infty}\frac{1}{n} {{2n-2}\choose {n-1}} x^n$$ How can I prove that?
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### How far is it from $A$ to $B$

If the boat moves downstream in the river, then going from country $A$ to country $B$ takes about $4$ hours, whereas if it moves upstream, then the same journey takes about $5$ hours. What is the ...
603 views

### Result of subtraction either positive or zero [closed]

I need to perform a subtraction where the result must be either positive or zero. I will use some pseudo code to represent what I mean (like it is not clear enough but still). ...
387 views

### Exponential Equations with Fractions

I have had some issues with the following two equations: $$\frac{3^{n-2}}{9^{1-n}}=9$$ $$\frac{5^{3n-3}}{25^{n-3}}=125$$ If anyone could work them out step by step that would be awesome. I ...
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### Solving an algebraic expression by taking common factors

First of all, I'd like to apologize for this question because I'm sure the answer is simple, I'm just a bit rusty on basic algebra skills after being out of practice for so long. I can't seem to work ...
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### Prove $\Sigma_{k=0}^{n-1}\lfloor x+\frac{k}{n}\rfloor=\lfloor nx\rfloor$ , n is a Natural Number

Prove the following identity: $$\lfloor x\rfloor +\lfloor x+\frac{1}{n}\rfloor +\lfloor x+\frac{2}{n}\rfloor +\lfloor x+\frac{3}{n}\rfloor+...+\lfloor x+\frac{n-1}{n}\rfloor =\lfloor nx\rfloor$$ ...
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### why cannot the limits be $-1$ and $-2$

I came across a problem in definite integral as : Evaluate $$I=\int_{0}^{3} x\sqrt{1+x}\:dx$$ By the substitution $1+x=t^2$ so book has given lower and upper limits as $t=1$ and $t=2$ which is ...
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### how many answers do this equation have$3^{2x}-34(15^{x-1})+5^{2x}=0$

How many answers do this equation have? $3^{2x}-34(15^{x-1})+5^{2x}=0$ My Attempt:$3^{2x}+5^{2x}=34(15^{x-1})$.Now what to do?
### Why is solution to this inequality equal to $\mathbb{R}$?
Given inequality $$-5(1- x)^2 < 3x + 11$$ which after algebraic manipulation looks like $$5x^2 - 7x + 16 > 0,$$ it is obvious that it's discriminant is equal to $\Delta = -271$. In my book it is ...