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.

47234 questions
4
votes
5 answers

Find the value of $3+7+12+18+25+\ldots=$

Now, this may be a very easy problem but I came across this in an examination and I could not solve it. Find the value of $$3+7+12+18+25+\ldots=$$ Now here is my…
Pole_Star
  • 1,082
4
votes
1 answer

How do you prove that an implicitly defined function cannot be made explicit?

Consider for example the function: $$x^2 e^y + \log(x)y^2 = 0$$ I suspect that neither x nor y can be isolated in this function (that it, it cannot be written as either $x(y)$ or $y(x)$. However, I can't really prove that other than to say "it's…
Michael Stachowsky
  • 2,943
4
votes
2 answers

Given $ax^2+by^2+cxy > 0$, what can I deduce about $a$, $b$, and $c$?

I know that for any nonzero $x,y\in\mathbf{R}$, $$ax^2+by^2+cxy > 0,$$ where $a,b,c\in\mathbf{R}$. What can I deduce about $a$, $b$, and $c$? For example, letting $x=1$ and $y=0$, I know that $\boxed{a>0}$. Letting $x=0$ and $y=1$, I know that…
Ron
  • 43
  • 2
4
votes
1 answer

How do I complete this square?

I need to simplify the following equation, by completing the square: $$\frac{\left(x-z\right)^{2}}{2(u-s)}+\frac{(z-y)^{2}}{2(t-u)}$$ As $\displaystyle\frac{\left(x-y\right)^{2}}{2(t-s)}+C$. How can I do this? I can't seem to be able to deal with…
squ1d
  • 139
4
votes
2 answers

How to show that $\left(\frac{y}{y-1}\right)^y = \left( 1 + \frac 1{y-1}\right)^y $

I need to show \[\left(\frac{y}{y-1}\right)^y = \left( 1 + \frac 1{y-1}\right)^y \] I cannot understand how they made this step! Can someone explain how this works?
Lukas Arvidsson
  • 2,376
4
votes
1 answer

Solve for $x$: $\big(x^3+\frac{1}{x^3}+1\big)^4=3\big(x^4+\frac{1}{x^4}+1\big)^3$

Solve for $x$ $$\big(x^3+\frac{1}{x^3}+1\big)^4=3\big(x^4+\frac{1}{x^4}+1\big)^3$$ let $x+\frac{1}{x}=t$ the equation equivalent to $(t^3-3t+1)^4=3(t^4-4t^2+3)^3$ but it's very complicated. Thanks.
tangkhaihanh
  • 917
4
votes
1 answer

Calculate PayPal's fees

I have a problem of ethics and mathematics. I have a website with subscription fees. My customers can pay them with PayPal. PayPal takes $3,4\%$ of my price plus a fixed fee of $0,25c$. If my customer pays $10$, PayPal takes $10 \times 0,034 + 0,25…
Raphaël
  • 143
4
votes
3 answers

Finding $\min \{(x + 6), (4 – x) \}$

Let $ y = \min \{ (x + 6), (4 – x) \}$, then find $y$. How to solve this problem?
user6868
4
votes
2 answers

Find the least number $x$ such that $ 11$ divides $x$ and sum of its digits $S(x)$ is $27$.

Find the least number $x$ such that $ 11$ divides $x$ and sum of its digits $S(x)$ is $27$. Since $S(999)=27 $ it is clear that the number of digits $n>3.$ Let $x_i$ be digits then we have two equations \begin{cases} x_1+x_2+\cdots+x_n=27=5 \mod…
Leox
  • 8,120
4
votes
1 answer

Proving the triangle inequality for the euclidean distance in the plane

I'm looking to introduce my students to the triangle inequality in the plane with the regular euclidean distance. They have no knowledge of functions or vectors (and therefore norms) so the proof should contain no mention of those concepts. I'm…
Lundborg
  • 1,646
4
votes
2 answers

How to use finite differences to determine an equation of a polynomial given consecutive integer $x$ and corresponding $y$ coordinates of the graph?

This chart is given: for $x=-3$, $y=-9$ for $x=-2$, $y=3$ for $x=-1$, $y=3$ for $x=0$, $y=-3$ for $x=1$, $y=-9$ for $x=2$, $y=-9$ for $x=3$, $y=3$ I found the finite differences to be 6 and degree of the polynomial with the given points to be 3…
Arthur Alex Karapetov
  • 409
4
votes
1 answer

A contractor had to finish a work in

A contractor had to finish a work in $40$ days and he employed some men to do the work. They finished one-fifth of the work in $20$ days. When $80$ more men were added, the work was finished on the specified time. How many men were employed in the…
pi-π
  • 7,416
4
votes
1 answer

Which number is bigger than the others?

Which number is bigger? $2^{431},3^{421},4^{321},21^{43},31^{42}$ My attempt: $4^{321}=2^{642}>2^{431},4^{321}=2^{642}>2^{640}=32^{128}>31^{42}$ $3^{421}>3^{420}=27^{140}>21^43$ But I don't know how to compare $4^{321}$ and $3^{421}$ Any hints?
Taha Akbari
  • 3,559
4
votes
1 answer

show that all the the roots of the equation $2x^3-ax^2+bx-c=0$ are real

let $k_{1}>k_{2}>k_{3}>k_{4}>k_{5}>k_{6}$ and $a=k_{1}+k_{2}+k_{3}+k_{4}+k_{5}+k_{6}$ and $b = k_{1}k_{3}+k_{3}k_{5}+k_{5}k_{1}+k_{2}k_{4}+k_{2}k_{6}+k_{4}k_{6}$ and $c=k_{1}k_{3}k_{5}+k_{2}k_{4}k_{6}$ show that all the the roots of the equation…
DXT
  • 11,241
4
votes
1 answer

Confusion over $a_k = (1+\frac{1}{k})^k = e$

I'm currently just watching an MIT lecture about differentiating exps and logs and it mentions that $a_k = (1+\frac{1}{k})^k = e.$ I've seen the proof and "understand" that $\lim_{k \to \infty} \ln(a_k) = 1$ s0 the limit of $a_k = e$. The problem is…
Ryan Tandy
  • 403
1 2 3
…
99 100