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.

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$\sqrt{x}$$\sqrt{x}$ = $x$ but $\sqrt{x^2}$ = $|x|$. Why?

$\sqrt{x}$$\sqrt{x}$ = $x$ but $\sqrt{x^2}$ = $|x|$. Why is this? I'm just learning algebra again after many years and I can't seem to figure out why this is. I'm sure this is trivial but if someone could explain it it would help me a lot. Thanks!
jon
  • 101
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Given 4 integers, $a, b, c, d > 0$, does $\frac{a}{b} < \frac{c}{d}$ imply $\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}$?

We were trying to come up with an easy way to generate a rational number in between two existing rational numbers with a fairly low numerator and denominator (the way we were doing this earlier was to find the average of the two rationals, but that…
Hans Z
  • 193
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If $x+\frac{1}{x}=\frac{1+\sqrt{5}}{2}$ then $x^{2000}+\frac{1}{x^{2000}}= $?

If $x+\frac{1}{x}=\frac{1+\sqrt{5}}{2}$ then $$x^{2000}+\frac{1}{x^{2000}}=?$$ My try: $$\left(x^{1000}\right)^2+\left(\frac{1}{x^{1000}}\right)^2=\left(x^{1000}+\frac{1}{x^{1000}}\right)^2-2$$ Continuation ?
zahra afshar
  • 153
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Factoring quadratics - how to?

It has been some time since I have studied math (and even then it was base-level math), and I know this will probably be easy for most, but I need help with Simplifying Algebra equations. Simple equations like so I can do: 3x + 7 + 9x -4 would (I…
DankyNanky
  • 193
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Finding polynomials with their values at points

Is there any way I can find a polynomial given any $2$ points (with $x$ coordinate OF MY CHOICE): Let's say there's some polynomial I don't know $(p(x)=2x^3+x^2+3)$, but my machine will give me an output. I give one $x$ value of my choice, and it…
bob the pie
  • 489
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If a,b,c are three distinct real numbers

If $a,b,c$ are three distinct real numbers and $$a+\frac1b=b+\frac1c=c+\frac1a=t$$ for some real number $t$ prove that $abc+t=0$
lokesh sangabattula
  • 657
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Confusing algebra rule: why $\frac{7^{n+1}-1}{6} + 7^{n+1} = \frac{7^{n+2}-1}{6}$?

Math rule I don't understand. My discrete math midterm is tomorrow and I'm studying proof styles. I came across a rule (algebra maybe?) I don't quite understand and I was hoping someone could explain it step by step for me. $$\frac{7^{n+1}-1}{6} +…
Irresponsible Newb
  • 1,323
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Can you solve this exponential equation?

I'm supposed to solve the equation $$3^x + 3^\sqrt x = 90$$ What steps do I need in order to get the solution $x=4$?
azazy
  • 91
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Ordered pairs of Integers $(x,y)$ which satisfy $x!\cdot y! = x!+y!+2$

Find the total number of ordered pairs $(x,y)\in\mathbb{Z}^2$ which satisfy $$ \tag1 x!y! = x!+y!+2 $$ My Attempt: We can write $(1)$ as $$ \begin{align} x!y!-x!-y!+1 &= 3\\ \left(x!-1\right)\left(y!-1\right) &= 3=1\times 3 = 3\times…
juantheron
  • 53,015
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Finding all possible values of $x^4+y^4+z^4$

Given real numbers $x,y,z$ satisfying $x+y+z=0$ and $$ \frac{x^4}{2x^2+yz}+\frac{y^4}{2y^2+zx}+\frac{z^4}{2z^2+xy}=1$$ Find all possible values of $x^4+y^4+z^4$ with proof. My attempt : Putting $x=-y-z$ and doing subsequent calculations we get…
shadow10
  • 5,616
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$p(x)=0$ with real coefficient has purely Imaginary roots.Then the equation $p(p(x)) = 0$ has

If the Quadratic equation $p(x)=0$ with real coefficient has purely Imaginary roots.Then the equation $p(p(x)) = 0$ has $\bf{OPTIONS::}$ $(a)\;\; $ Only purely Inaginary Roots. $\;\;\;\;\;\;(b)$ all real roots. $(c)$ Two real and Two purely…
juantheron
  • 53,015
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If $\dfrac{a}{b-c}+\dfrac{b}{c-a}+\dfrac{c}{a-b}=0$ with unequal $a,b,c$, Prove that $\dfrac{a}{(b-c)^2}+\dfrac{b}{(c-a)^2}+\dfrac{c}{(a-b)^2}=0$

If $\dfrac{a}{b-c}+\dfrac{b}{c-a}+\dfrac{c}{a-b}=0$ with unequal $a,b,c$, Prove that $\dfrac{a}{(b-c)^2}+\dfrac{b}{(c-a)^2}+\dfrac{c}{(a-b)^2}=0$ I could not approach the problem at all though I think I could have done something by using…
Hawk
  • 6,540
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1 answer

Summation - relatively simple?

I have a question which might be too simple for this site but I really tried many ideas without coming to a solution. This is assignment from elementary school in which I am trying to help and the solution should be relatively simple but somehow I…
user120250
  • 377
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7 answers

Prove no real number satisfies $x^{2} = -1$

I ran a search, but, oddly enough, I can't to find a similar question on here. (If so, kindly point me in that direction, and I'll take this one down.) It seems like a pretty basic question in real analysis, but I'm struggling to come up with a…
user96966
  • 103
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Proof of identity $\sqrt {xy} = \sqrt x \sqrt y$ for $x,y \in \mathbb R^+$

Proof of identity $\sqrt {xy} = \sqrt x \sqrt y$ for $x,y \in \mathbb R^+$ I've been looking at the stated identity, which makes sense in $\mathbb R^+$ but fails in $\mathbb R$, since $\sqrt {-1 \cdot -1} \neq \sqrt {-1} \sqrt {-1}$. How does one…
Shuzheng
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