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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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If $(b-a) = (ba^2 - ab^2)$, show that $a=b$ where $a,b \in \mathbb R$

I am slightly stuck on this seemlingly simple problem that I encoutered as part of a problem to show that the orthogonality condition of $M_{2\times2}$ matrices given by $\sum_i a_{ij}a_{jk} = \delta_{jk}$, where the $a_{ij}$ are the matrix entries,…
Floris
  • 139
4
votes
3 answers

Find $x$ if $(7+4\sqrt 3)^{x^2-8}+(7-4\sqrt 3)^{x^2-8}=14$

Known that: $$(7+4\sqrt 3)^{x^2-8}+(7-4\sqrt 3)^{x^2-8}=14$$ What is the value of $x$
program-o-steve
  • 93
4
votes
8 answers

How to solve $x$ in $(x+1)^4+(x-1)^4=16$?

I'm trying my hand on these types of expressions. How to solve $x$ in $(x+1)^4+(x-1)^4=16$? Please write any idea you have, and try to keep it simple. Thanks.
jackcall
  • 83
4
votes
3 answers

Two Questions Regarding Polynomial Functions.

I am studying Polynomial functions and their Graphs. I am currently looking at the definition for a polynomial function and I am trying to arrive at a deeper understanding; thus, please excuse questions that seem obvious. Nonetheless, A polynomial…
Prologue
  • 235
4
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1 answer

Steel mixture homework problem

My problem is the following: In a particular steel mixture, $20 \%$ of it consists from chromium and $10\%$ from nickel. How much of this steel mixture can be produced, if we have $1200$kg of chromium and $900$kg of nickel. The other steel…
jjepsuomi
  • 8,619
4
votes
4 answers

What does $f(x)$ denote?

All I have been reading from my textbook, and I am wondering if my textbook is just badly written since I have learned in the past that $f(x)$ means $y$ (output) and the variable $x$ is the input. However, the book is using $f(x)$ as the expression.…
Cetshwayo
  • 3,092
4
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3 answers

How do I evaluate $\log_3(x^2 − 36) − \log_3(x − 6)?$

The questions ask to combine the expression using the laws of logarithms. $$\log_3(x^2 − 36) − \log_3(x − 6)$$ I got $$\frac{\log_3(x^2-36)}{\log_3(x-6) } $$ but my book says that is not correct.
Frustrated with Math
  • 131
4
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1 answer

Solving a system of 3 quadratic simultaneous equations.

$x, y, z$ are positive real numbers which satisfy the following three equations. $$x + 2y + z = 5(x + y)(y + z)$$ $$x + y + 2z = 7(y + z)(z + x)$$ $$2x + y + z = 6(z + x)(x + y)$$ Find the value of $(24)^3(xyz)$. Okay , so here I'm probably not…
A Googler
  • 3,355
4
votes
2 answers

Is ma solution correct to this equation?

$$\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}=3$$ My take: $$\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}=3\implies {x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}=9$$ Thus: $$x+3=9\implies x=6$$ Did I do it right?
user137465
  • 43
4
votes
2 answers

Show that $e^x + e^{-x} =2 +x^2$

I have a little problem with the following question: Given that $e^x$ is approximately equal to $1+x+\frac{x^2}{2}$. Show that $e^x + e^{-x} = 2 + x^2$ I can do this by simply long dividing $$\frac{(1+x+\frac{x^2}{2})^2…
user
  • 938
4
votes
4 answers

Why does $\frac{1}{\sin x} = 2\sin x$?

I'm trying to understand the solution of a trigonometry problem. One of the steps of the solution says that: $$\frac{\sqrt2}{2} = \sin x$$ And then directly deduces that: $$\sqrt2 = \frac{1}{\sin x}$$ I wonder how this equivalence works. It looks…
Alex Marandon
  • 193
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votes
0 answers

find all natural $n$ such that equation is a sixth power

We have $24n^3+6n+7$ and we need to find all natural $n$ for which this equation is a sixth power of natural number. My try: let $k \in N$ $24n^3+6n+7=k^6$ $24n^3+6n+6=k^6-1$ $6(2n+1)(2n^2-n+1)=(k^3+1)(k^3-1)$ let…
Gregor
  • 708
4
votes
3 answers

How can you use algebra to show that $a^2 - b^2 = (a+b)(a-b)?$

It's easy enough to show that $(a+b)(a-b) = a^2 - b^2$, you just simply multiply it out: $$(a + b)(a - b)$$ $$= a^2 -ab + ba - b^2$$ $$= a^2 - b^2$$ However, is there any way using basic algebra that you can go from $a^2 - b^2$ to $(a + b)(a - b)$?…
Daniel T.
  • 143
4
votes
1 answer

Exponential scaling?

I have a series of numbers, say 2, 3, 7 that sum 12. I want to increase these numbers so that they sum 62. I could multiply each number by 62/12 to achieve that. However, all numbers would be scaled equally linearly. I want instead that 7 be scaled…
alex
  • 195
4
votes
3 answers

Absolute inequality question.

I'm stuck with this kind of absolute inequality: $$|x+1|>x+2. \tag1$$ Firstly, when I solve this one: $$|x+1|=x+2, \tag2$$ I make sure the right side of the equation is greater than zero; Condition: $$x+2\ge0;\quad x\ge-2.$$ When I solve this by…
Guest
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