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
10
votes
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Proof that $\frac{1}{a_1} +\frac{1}{a_2} +...+\frac{1}{a_{20}}$ is an integer

Assume that, for $n\ge1$,$$a_n=\sqrt{1+\left(1+\frac{1}{n}\right)^2 } +\sqrt{1+\left(1-\frac{1}{n}\right)^2 } $$ How to prove that $$\frac{1}{a_1} +\frac{1}{a_2} +...+\frac{1}{a_{20}}$$ is an integer?
kun
  • 101
10
votes
1 answer

Calculate "x % slower/faster"?

Ok, this might sound a bit stupid, but I find these kind of statements "x % slower/faster" pretty confusing. Let's say I have algorithm A and algorithm B. Algorithm A takes 50 seconds to complete a task and algorithm B takes 25 seconds. Now I could…
tsauerwein
  • 201
10
votes
3 answers

Why is finding the roots of a polynomial equation so important? What is to gain?

I have just started a pre calculus class, and our first lessons have been reviews on polynomial equation, quadratics and finding roots or solutions to equations. The topic is fairly simple but I just have to know why is it so important to find an X…
hubble
  • 619
10
votes
1 answer

Cyclic system of cubic equations in $5$ variables

Problem: To find all real solutions of the system: $$3a=(b+c+d)^3$$ $$3b=(c+d+e)^3$$ $$3c=(d+e+a)^3$$ $$3d=(e+a+b)^3$$ $$3e=(a+b+c)^3$$ My attempt: I tried to get a bound for positive solutions. Using AM$\geq$GM, $(x+y+z)^3\geq27xyz$, I get $$…
Potemkin Metro Card
  • 1,598
10
votes
11 answers

Solving $5^n > 4,000,000$ without a calculator

If $n$ is an integer and $5^n > 4,000,000.$ What is the least possible value of $n$? (answer: $10$) How could I find the value of $n$ without using a calculator ?
MistyD
  • 1,655
10
votes
2 answers

Solving $x^2+\frac{81x^2}{(9+x)^2}=40$

Solve the following equation: $$x^2+\dfrac{81x^2}{(9+x)^2}=40$$ Unfortunately I have no ideas because after expanding I get an equation of 4 degree.
RFZ
  • 16,814
10
votes
2 answers

How do you find a vector in the form when only the angle and magnitude are given?

How do you find a vector in the form when only the angle and magnitude are given? Here is an example where an angle of 80 degrees is given along with a magnitude of 3.
Curious
  • 467
10
votes
4 answers

How can one solve the equation $\sqrt{x\sqrt{x} - x} = 1-x$?

$$\sqrt{x\sqrt{x} - x} = 1-x$$ I know the solution but have no idea how to solve it analytically.
user1111261
  • 1,149
10
votes
2 answers

How can we simplify the expression $P+\sqrt{P^2+\sqrt{P^4+\sqrt{ P^8+\cdots)}}}$?

Is there a way to reduce the expression $P+\sqrt{P^2+\sqrt{P^4+\sqrt{ P^8+\cdots)}}}$?
Sumit Jha
  • 153
9
votes
4 answers

Find the last non zero digit of 28!.

Find the last non zero digit of 28!. It is very hard to multiply and find the last nonzero digit. I just wanna know that, is there any easy technique to solve this type of problem?
Numerical Person
  • 977
9
votes
4 answers

if $x+y+z=2$ and $xy+yz+zx=1$,Prove $x,y,z \in \left [0,\frac{4}{3} \right ]$

if $x+y+z=2$ and $xy+yz+zx=1$,Prove $x,y,z \in \left [0,\frac{4}{3} \right ]$ things i have done: first thing to do is to show that $x,y,z$ are non-negative. $$xy+yz+zx=1 \Rightarrow zx=1-yz-xy \Rightarrow zx=1-y(z+x)\Rightarrow…
user2838619
  • 3,120
9
votes
2 answers

Real solution of the equation $(x^2-2x+2)^2-2(x^2-2x+2)+2 = x$

Calculate all real solutions $x\in\mathbb{R}$ of the equation $$ \tag1(x^2-2x+2)^2-2(x^2-2x+2)+2 = x $$ My Attempt: I used the concept of a composite function. Let $f(x) = x^2-2x+2$. Then equation $(1)$ converts into $f(f(x)) = x$. Both $f(x) =…
juantheron
  • 53,015
9
votes
4 answers

Number of digits of the number of digits of the number of digits of $2014^{2014}$

How would you solve that problem : What is the number of digits of the number of digits of the number of digits of $2014^{2014}$ ? (for instance the number of digits of $12345678901234567890$ is $20$, and the numbers of digits of $20$ is $2$, and…
Hippalectryon
  • 7,730
9
votes
4 answers

Solving $a^{x} = 10^{2x + 1}$

So here is the problem: Solved for a in terms of x: $$a^{x} = 10^{2x + 1}$$ I tried: $\displaystyle x \cdot \log(a) = (2x+1) \cdot \log\;10 $ $\displaystyle \frac{x}{2x + 1} = \frac{\log\;10} {\log\;a} $ But this is not going in the right…
gideon
  • 403
9
votes
3 answers

Factor $x^4 + 64$

I found a page where this problem was solved but his english is broken so its difficult to understand his explanation. His first step was to divide the constant, 64, by the exponent 4. What is his reasoning behind this…
user135247
  • 207
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