Mathematics Stack Exchange
Mathematics Stack Exchange

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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Domain when dividing rational expression

I'm having issue understanding the process of defining a domain while attempting to divide rational expressions: $$ \frac {x^2+x-6}{x^2+3x-10} : \frac {x+3}{x-5} $$ We can factor to the form $$ \frac {(x+3)(x-2)}{(x+5)(x-2)} : \frac {(x+3)}{(x-5)}…
Jakub
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Evaluating $\frac{2013^3-2\cdot 2013^2\cdot 2014+3\cdot 2013\cdot 2014^2-2014^3+1}{2013\cdot 2014}$

What is the value of $$\frac{2013^3-2\cdot 2013^2\cdot 2014+3\cdot 2013\cdot 2014^2-2014^3+1}{2013\cdot 2014}?$$ What I have tried: $$\implies\frac{2013^2(2013-2\cdot2014)+2014^2(3\cdot 2013-2014)+1}{2013\cdot…
Max0815
  • 3,505
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Solving equations involving square roots

I am a student and I often encounter these type of equations: $$\sqrt{x^2 + (y-2)^2} + \sqrt{x^2 + (y+2)^2} = 6$$ I usually solve these by taking one term ($\sqrt{x^2 + (y-2)^2}$ for example) to the right hand side but this seems to take more time.…
Shantanu Kaushik
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Calculation of polynomial $g(x)$ satisfies $x\cdot g(x+1)=(x-3)\cdot g(x)$

If a polynomial $g(x)$ satisfies $x\cdot g(x+1)=(x-3)\cdot g(x)$ for all $x$, and $g(3)=6$, then $g(25)=$? My try: $x\cdot g(x+1)=(x-3)\cdot g(x)$, Put $x=3$, we get $g(4)=0$, means $(x-4)$ is a factor of $g(x)$. Similarly put $x=0$. We get…
juantheron
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Simplify $\sqrt{\dfrac{\sqrt[3]{64} + \sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}}$ into $\dfrac{\sqrt{3}}{3}$

I am on the final question of a textbook chapter on radicals and this question feels more challenging, perhaps that's the idea. If you view my post history I typically make a effort to provide some working to simplify the expression to an extent,…
Doug Fir
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Manipulating this $\frac{x-y}{z-y}$ to $1+\frac{x-z}{z-y}$

There is probably a very easy explanation for this that is lost on me. Came across a formula that was manipulated into another form and it was presented as a given, so I am trying to figure out how that was…
TheCuriousOne
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Find all integer solutions for $2(x^2+y^2)+x+y=5xy$

Find all integer solutions for $2(x^2+y^2)+x+y=5xy$ I have been attempting to solve this question for a long period of time but have never achieved anything. I tried to go back to WolframAlpha and it gave me that the integer solutions were $x=y=2,…
user587054
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Why do I keep getting this incorrect solution when trying to find all the real solutions for $\sqrt{2x-3}\ +x=3$.

The problem is to find all real solutions (if any exists) for $\sqrt{2x-3}\ +x=3$. Now, my textbook says the answer is {2}, however, I keep getting {2, 6}. I've tried multiple approaches, but here is one of them: I got rid of the root by squaring…
Lex_i
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What's the smallest positive integer made of 1's that is divisible by a number made out of 100 9's

I was asked what is the smallest positive integer made of ones (11111...1) that is divisible (no reminder) by a number made up of 100 digits of 9 (9999...9). I noticed that for 9 the smallest integer made out of ones that will divisible by him, is…
user9014
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$(\frac1{n^{\sqrt n}})^{\frac1n}=(n^{-\frac{\sqrt n}{n}})^{}$

I'm a little bit confused by this one. Is this correct? $$\left(\frac1{n^{\sqrt n}}\right)^{\frac1n}=\left(n^{-\frac{\sqrt n}{n}}\right)^{}=\sqrt{n^{-\frac1n}}$$ ${}{}{}{}{}{}{}{}{}{}{}{}$ Edit: Is it okay that I changed the question a little bit ?
Kasper
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What is rule for when solving algebraic equations?

I'm a high school student trying to get critical intuition when learning algebraic equation solving. For $x$ any complex number and $c$ constant, simple polynomial such as $x^n -c=0$ are easily solvable for $x$. Then if we know how to solve…
Solvable Potato
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Why would I divide by 4 instead of 2 in this equation?

So I have this problem: A vertical flag pole of height $h\;\text{meters}$ is erected exactly in the middle of the flat roof of a building. The roof is rectangular of width $w\;\text{meters}$ and depth $d\;\text{meters}$. The flag pole is stabilized…
Ballusha Three
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write every integer as a signed sum of integer power an integer.

This has been an issue, I haven't been able to solve it yet despite many attempts. The problem is the following: $ \forall n \in \mathbb{N}^*, \forall M \in \mathbb{N}, \exists K \in \mathbb{N} | \exists (\epsilon_i)_{0
Frayal
  • 299
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$a(x-a)^2+b(x-b)^2=0$ has one solution; $a, b$ are not $0$. Prove $|a|=|b|$

$a(x-a)^2+b(x-b)^2=0$ has one solution; $a, b$ are not $0$. Prove $|a|=|b|$ simplifying the equation I got: $$(a+b)x^2-2(a^2+b^2)x+(a^3+b^3)=0$$ solving for the Discriminant $D=0$, I got: $$a(a^2b-2ab^2+b^3)=0$$ and since $a$ cannot equal $0$, I…
Pero
  • 289
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Given $n$ integers, is it always possible to choose $m$ from them so that their sum is a multiple of $m$?

The original question: given $6666$ integers, (positive, negative or $0$) is it always possible to choose $2018$ from them so that the chosen numbers add to a multiple of $2018$? (positive multiple, negative multiple or $0$) Prove or disprove. More…
L. F.
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