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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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Number of divisors of $9!$ which are of the form $3m+2$

Total number of divisors of $9!$ which are is in the form of $3m+2$, where $m\in \mathbb{N}$ My Try: Let $ N = 9! = 1\times 2 \times 3 \times 2^2 \times 5 \times 2 \times 3 \times 7 \times 2^3 \times 3^2 = 2^7 \times 3^4 \times 5 \times 7$ Now If…
juantheron
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$2d^2=n^2$ implies that $n$ is multiple of 2

I'm reading a proof of the irrationality of $\sqrt 2$. In a step it states that $2d^2=n^2$ implies that $n$ is multiple of $2$. How?
bigstones
  • 211
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Find the domain of $\sqrt{x^2-9}$

$$\sqrt{x^2-9}$$ I know that the domain of square root is greater than or equal to zero. I solve for when $x^2-9<0$ and get $x^2<9$. Now I get $x<-3$ and $x<3$. I know that the domain is $(-\infty,-3] \cup [3,\infty)$ The problem is I do not…
Kot
  • 3,273
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Solving this problem algebraically: $81^{\sin^{2}(x)} + 81^{\cos^{2}(x)} = 30 $

I saw this problem on an instagram feed $$81^{\sin^{2}(x)} + 81^{\cos^{2}(x)} = 30$$ solve for x And I saw that most people had solutions where you plug in x = 30 degrees or $\frac{\pi}{6}$ and the like. I get that if you assume that the two numbers…
Jesse
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How to prove $x + 3 = (x + 1)^2 \Leftarrow x=-2\;\text{or}\;1 \,? \quad \sqrt{x+3}=x+1 \Leftarrow x=1 \, ? \quad x^2 = x \Leftarrow x=0 \; $?

I can't deduce the $\color{blue}\impliedby$ of the 3 blue biconditionals below. How do I explicitly wrest or wring out the red functions in terms of x, from the 1 or 2 integers on the RHS? Doubtless, I know that the RHS's roots satisfy the LHS! …
user1124753
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On the topic of Trinomial Expansion

So I was looking at the Wikipedia of the binomial theorem when I read this: Multi-binomial theorem When working in more than two dimensions, it is often useful to deal with products of binomial expansions. By the binomial theorem this is equal…
CrSb0001
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Population of a city doubles in $50$ years. In how many years will it triple?

The population of a city doubles in $50$ years. In how many years will it triple, under the assumption that the rate of increase is proportional to the number of inhabitants? This was a pretty trivial question. But, the thing is that when I tried…
Thomas Finley
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Find the product of $5x^2+5xy-9y^2$ and $5x^2-5xy-9y^2$

Find the product of $5x^2+5xy-9y^2$ and $5x^2-5xy-9y^2$ I did a straight-up multiplication of every term and got the correct answer, $25x^4-115x^2y^2+81y^4$ However given the identical expressions given (differing only in sign), I can't help but…
ronald christenkkson
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Distributive Property Theory question

I've been wondering about the theory behind the distributive property lately. For example: 2(pi * r^2) is just 2 * pi * r^2. However, when you add a positive number like +3. You get 2(PI*r^2 +3). But, that isn't just 2*PI*r^2+3. Its: 2pi*r^2 +…
user2608474
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Value of expression $(\alpha^{110}+\beta^{110})-(\alpha^{98}+\beta^{98})$ in given quadratic equation

If $\alpha,\beta$ are the roots of the equation $x^2+3^{\frac{1}{4}}x+3^{\frac{1}{2}}=0$. Then value of $\displaystyle \alpha^{98}(\alpha^{12}-1)+\beta^{98}(\beta^{12}-1)$ is From $\displaystyle x^2+3^{\frac{1}{4}}x+3^{\frac{1}{2}}=0$,we…
Priti Bisht
  • 421
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Solve $\frac{x-8}{x-10}+\frac{x-4}{x-6}=\frac{x-5}{x-7}+\frac{x-7}{x-9}$

Solve $\frac{x-8}{x-10}+\frac{x-4}{x-6}=\frac{x-5}{x-7}+\frac{x-7}{x-9}$ $\Rightarrow \frac{(x-10)+2}{x-10}+\frac{(x-6)+2}{x-6}=\frac{(x-7)+2}{x-7}+\frac{(x-9)+2}{x-9} \ \ \ ...(1)$ $\Rightarrow…
ronald christenkkson
  • 1,623
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5 answers

Finding $x-\frac{1}{x}$, given $x^3 - \frac{1}{x^3} = 108+76\sqrt{2}$

If $x^3 - \dfrac{1}{x^3} = 108+76\sqrt{2}$, find the value of $x-\dfrac{1}{x}$. Here's what I've tried so far. $$\begin{align} \left(x-\dfrac{1}{x}\right)^3&=x^3-\dfrac{1}{x^3}-3\left(x-\dfrac{1}{x}\right) \\ \rightarrow \quad…
avighnac
  • 355
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1 answer

Proving that an algebraic expression cannot be a square

Say that I have an expression in several variables, like $zxy+z^5x^2y^2 + xy + 24z$. To prove that it's not a perfect square, I write it in terms of one of the variables, say $x$. This makes it a polynomial of degree $2$. Now expand out $(a+bx)^2$…
user75122
  • 293
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Is my work transforming $9(x+2)^2$ to $(3x+6)^2$ correct? What is this method called?

$$9(x+2)^2 = 3^2(x+2)^2=(3(x+2))^2=(3x+6)^2$$ I want to know if the use of brackets in this problem has been done correctly. What is this method called?
ilove cupcakes
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How many different number are there in sequence $⌊n/1\rfloor$, $⌊n/2⌋, ⌊n/3⌋, ..., ⌊n/n]$?

$1 \leq n \leq 10^{12}$ and n is an integer. Through some documents on the internet, I vaguely know that the answer is $\sqrt{n}$ or maybe $2\sqrt{(n)}$. Can anyone help me to prove this? My English is not very good, sorry if it makes you confused.
qvanle
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