# 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.

32,603 questions
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### How to find the smallest side of a triangle when the interior angles are unknown?

I am confused on how to find the answer for this problem. So far what I believe would apply is the triangle inequality but I'm not sure on how to use it. The figure $ABC$ is a triangle so as $BDC$. ...
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### Sum of all real numbers $x$ such that $(\text{A quadratic})^\text{Another quadratic}=1$.

What is the sum of all real numbers $x$ such that $(x^2-5x+5)^{(x^2-7x+12)}=1$? So I know that $x^0=1$ and $1^x=1$. So, I can solve for them and find $x$, and add them up. Solving $x^2-7x+12=0$ ...
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### Methods for finding range of a function

I'm aware that this can't be done in general, and that this is a broad topic, but are there some fast, applicable methods for finding range of a function ? Can we use intermediate value theorem, and ...
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### Algebra problem asking for the sum of two people's ages.

A very simple problem tends to become very hard. Perhaps I am overthinking it. If the square of Winslow's age is added to Abby's age, the sum is 209. If the square of Abby's age is added to ...
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### Summation formula for basic math question

I am reviewing some old retired case studies we have at work that we used to give to candidates interviewing. There is a short question on one part of the case that is pretty straight forward, but I ...
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### Values of $x$ satisfying $\lfloor{x^2\rfloor}=\left(\lfloor{x\rfloor}\right)^2$

Prove that values of $x$ satisfying $\lfloor{x^2\rfloor}=\left(\lfloor{x\rfloor}\right)^2$ is $(0,\sqrt{2})\cup \mathbb{Z}$ My try: Its trivial that every integer satisfies the given equation. Now ...
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### How do I get from log F = log G + log m - log(1/M) - 2 log r to a solution withoug logs?

I've been self-studying from Stroud & Booth's excellent "Engineering Mathematics", and am currently on the "Algebra" section. I understand everything pretty well, except when it comes to the ...
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### System of three non linear equations with three unknowns with random coefficients

I have the system of equations: $$\begin{cases} Ax + By + Cz &= D \\ Exy + Fxz + Gyz &= H \\ Ixyz &= J \\ \end{cases}$$ Where $A,B,C,D,E,F,G,H,I,J$ are constant integers between 1 and 9. ...
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### Is it possible to create an index relating two deviations but preserving the signal of one?

I have three temperatures I would like to compare. One is the body temperature ($T_{b}$), another is the reference ($T_{ref}$) and a third is the environment ($T_e$). I would like to create an index ...
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### continuity of function at one point [on hold]

We assume that the function $f(x)$ is defined by $$f(x) = \left\{ \begin{array}{cc} x+1, & x\leq 0 \\ -\frac{1}{2}x + 7, & x > 0 \end{array} \right.$$ Find a real number such that the ...
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### Newton's Generalization of the Binomial Theorem and big o notation

we know that : (Newton's Generalization of the Binomial Theorem) Let $x,y∈\mathbb{R}$ where $0≤∣x∣<∣y∣$ and let $α∈\mathbb{R}$. Then the expansion of the binomial $(x+y)^α$ is given by the ...
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### Prove or disprove: If $A$ is $n\times n$ and $\exists\;m\in \Bbb{N}:\;A^m=I_n$, then $A$ is invertible.
Is this statement true? If $A$ is an $n\times n$ matrix and $A^m=I_n$ for some $m\in \Bbb{N}$, then $A$ is invertible. My trial Let $n\in \Bbb{N}$ be fixed. Then, $$[\det(A)]^m=\det(A^m)=I_n=1.$$ ...
So I have a question like this. Given that $x + 1/y = 3/2$ and $y + 1/x = 1/6$, what is the value of $x / y$? So basically I'm trying to get $x$ and $y$ on their own but I don't know how to because ...