Mathematics Stack Exchange
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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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A real solution to a cubic equation

What is the easiest way to find the real solution of the equation $x^3-6x^2+6x-2=0$? I know the solution to be $x=2+2^{2/3}+2^{1/3}$ (Mathematica) but I would like to find it analytically. If possible, not by plugging the coefficients in Cardano's…
user54031
14
votes
6 answers

Typo in math brainteaser book?

A steamboat leaving pier $1$ takes $20$ hours to go against the current upriver to pier $2$. It can return downriver with the current from pier $2$ to pier $1$ in $15$ hours. If there were no current, how long would it take for the steamboat to…
Mick
  • 151
14
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6 answers

Average of all 6 digit numbers that contain only digits $1,2,3,4,5$

How do I find the average of all $6$ digit numbers which consist of only digits $1,2,3,4$ and $5$? Do I have to list all the possible numbers and then divide the sum by the count? There has to be a more efficient way, right? Thank you!
user88999
  • 141
14
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3 answers

Solution of $ax=a^x$

When I studied math at my school (I was 16) I was studying system of linear equations, and equations of first or second order and I liked it very much (and I was good to xD) but one day I asked to myself how to solve equation of this kind $ax=a^x$…
MphLee
  • 2,472
14
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4 answers

Simplifying $\sqrt{5+2\sqrt{5+2\sqrt{5+2\sqrt {5 +\cdots}}}}$

How to simplify the expression: $$\sqrt{5+2\sqrt{5+2\sqrt{5+2\sqrt{\cdots}}}}.$$ If I could at least know what kind of reference there is that would explain these type of expressions that would be very helpful. Thank you.
hyg17
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14
votes
3 answers

Counting number of students who have failed in all four subjects

In an examination of a certain class, at least $69\%$ of the students failed in P, at least $72\%$ failed in C, at least $80\%$ failed in M, and at least $85\%$ failed in B. How many at least must have failed in all the four subjects? My…
Gaurang Tandon
  • 6,449
13
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4 answers

How to solve $e^x=x$?

I mean how to solve $e^x=x$ in complex plane? Is there a solution? I know there is no real solution to this because apparently y=x and y=$e^x$ have no interception.
Xiangru Lian
  • 367
13
votes
5 answers

If $K = \frac{2}{1}\times \frac{4}{3}\times \cdots \times \frac{100}{99}.$ Then value of $\lfloor K \rfloor$

Let $K = \frac{2}{1}\times \frac{4}{3}\times \frac{6}{5}\times \frac{8}{7}\times \cdots \times \frac{100}{99}.$ Then what is the value of $\lfloor K \rfloor$, where $\lfloor x \rfloor$ is the floor function? My Attempt: By factoring out powers of…
juantheron
  • 53,015
13
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4 answers

$ \Big(\dfrac{x^7+y^7+z^7}{7}\Big)^2=\Big(\dfrac{x^5+y^5+z^5}{5}\Big)^2\cdot\Big(\dfrac{x^4+y^4+z^4}{2}\Big) $

I have a question. I tried so to solve it, but there is a problem. that is i don't have any idea to findout how can i work with degrees 4,5,7 ... this is the problem : let $ x , y $ and $ z $ three real numbers such $ x+y+z = 0 $. prove : $…
Master.AKA
  • 995
13
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5 answers

How can I write an equation that matches any sequence?

One thing I have been wondering about lately is how to write an equation that describes a pattern of numbers. What I mean is: x 0 1 2 y 1 5 9 If I have this, I can tell that an equation that describes this would be $y=4x+1$. In…
Larry Wang
  • 9,513
13
votes
4 answers

Prove maximum value of $(z-xy)(x-yz)(y-zx)$ is $\frac{1}{64}$ given $x,y,z \in (0,1)$

Prove maximum value of $(z-xy)(x-yz)(y-zx)$ is $\frac{1}{64}$ given $x,y,z \in (0,1)$ I can make it $\frac{1}{64}$ by setting $x,y,z = \frac{1}{2}$, but I have no idea how to show that's the maximum.
stevemarvell
  • 679
13
votes
9 answers

How to solve for $x$ in the equation, $4\sqrt{x-3} - \sqrt{6x-17} = 3$, with two square root terms?

So guys, my girlfriend is taking a college algebra class this summer and I figured I would help her study for her upcoming final because I am an engineering major and this kind of math would be easy for me. As we were doing problems, we came about…
Greg Harrington
  • 565
13
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7 answers

Why do we sometimes lose solutions when solving equations?

$\forall f:x=y \implies f(x)=f(y)$ which means that any operation can be done on both sides of an equation. When we solve equations we do one operation after the other: $$ x=y \implies f_1(x)=f_1(y) \implies f_2(f_1(x))=f_2(f_1(y)) \implies \dots…
mawaior
  • 375
13
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4 answers

How to solve : $\,8^x=6x$

I am stuck on the following problem which one of my friends gave me: Solve : $\,8^x=6x$. MY ATTEMPTS: We see that $$8^x=6x \implies 2^{3x}=6x.$$ Now I am not sure how to proceed further. Taking logarithm on both sides of the equation does not…
learner
  • 6,726
13
votes
3 answers

How to simplify $a^n - b^n$?

How to simplify $a^n - b^n$? If it would be $(a+b)^n$, then I could use the binomial theorem, but it's a bit different, and I have no idea how to solve it. Thanks in advance.
Element Scope
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