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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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If $(w + 1)(w - 1) = w$, find $ { w }^{ 10 }+\frac { 1 }{ { w }^{ 10 } } $.

Recently I was asked a question by my student that completely stumped me. $$\text{If }(w + 1)(w - 1) = w\text{, find } { w }^{ 10 }+\frac { 1 }{ { w }^{ 10 } }. $$ One "cheat" method that I used was to solve for the exact value of $w$ from the…
Eric
  • 71
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$x^4 - 4x^3 + 6x^2 - 4x + 1 = 0$

Determine all the possibilities for rational roots of the polynomial $x^4 - 4x^3 + 6x^2 - 4x + 1 = 0$. Then determine how many of the real roots of the polynomial may be positive and how many may be negative. Factor the polynomial to confirm your…
William Zlacki
  • 598
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1 answer

The square of a infinite series.

Say we have $\sum_{n=1}^{\infty} f(n,x)=f(x)$, which often happens with Taylor series: Can we express: $$\left(\sum_{n=1}^{\infty} f(n,x) \right)^2$$ As something that does not involve the square. I.e can multiply this out…
Ahmed S. Attaalla
  • 18,518
7
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4 answers

3xy + 14x + 17y + 71 = 0 need some advice

$$3xy + 14x + 17y + 71 = 0$$ Need to find both $x$ and $y$. If there was only one variable then this is easy problem. Have tried: $$\begin{align}3xy &= -14x - 17y - 71 \\ x &= \frac{-14x - 17y - 71}{3y}\end{align}$$ Then tried to put this expression…
Yang
  • 185
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4 answers

Does $\sqrt{\cos{\theta} \sqrt{\cos{\theta} \sqrt{\cos{\theta\dots}}}}=\sqrt{1 \sqrt{1 \sqrt{1\dots}}} \implies \cos{\theta}=1$?

I was solving this equation: $$\sqrt{\cos{\theta} \sqrt{\cos{\theta} \sqrt{\cos{\theta\dots}}}}=1$$ I solved it like this: The given equation can be written as: \begin{align*} \sqrt{\cos{\theta} \sqrt{\cos{\theta} \sqrt{\cos{\theta\dots}}}}&=\sqrt{1…
Man_Of_Wisdom
  • 649
7
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2 answers

Triples of positive real numbers $(a,b,c)$ such that $\lfloor a\rfloor bc=3,\; a\lfloor b\rfloor c=4,\;ab\lfloor c\rfloor=5$

Find the all ordered triplets of positive real numbers $(a,b,c)$ such that: $$\lfloor a\rfloor bc=3,\quad a\lfloor b\rfloor c=4,\quad ab\lfloor c\rfloor=5,$$ where $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$.
Frank
  • 2,738
7
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3 answers

How to solve this equation manually: $(x^2+100)^2=(x^3-100)^3$?

Well, I was given a problem, find $x$, if: $$(x^2+100)^2=(x^3-100)^3$$ I tried everything that I could, I even opened up the brackets which gave an ugly degree 9 equation, I also tried to plot the curves $y=\left(x^2+100\right)^2$ and…
Nikunj
  • 6,160
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A formula which gives the maximum of a series of numbers

This formula gives the maximum of 3 numbers: $$\frac{a}{2} + \frac{b}{4} + \frac{c}{4} + \frac{|b-c|}{4} + \frac{1}{2}\left|a -\frac{b}{2} - \frac{c}{2} - \frac{|b-c|}{2}\right| = \max(a,b,c)$$ I've found this over the internet, I have no idea how…
aqww
  • 177
7
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3 answers

How find the sum of the last two digits of $(x^{2})^{2013} + \frac{1}{(x^{2})^{2013}}$ for $x + \frac{1}{x} = 3$?

Let x be a real number so that $x + \frac{1}{x} = 3$. How find the sum of the last two digits of $(x^{2})^{2013} + \frac{1}{(x^{2})^{2013}}$?
piteer
  • 6,310
7
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2 answers

Prove that $\sqrt[3]5 - \sqrt[4]3$ is Irrational

I've gone many directions and they all fail. The sum of two irrationals doesn't need to be irrational. I found a proof saying: if irrational $x,y$ have a rational sum $x+y$, then $x-y$ is irrational, or vice versa. However, in this case $x+y$ and…
Julian73
  • 149
7
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2 answers

Formula for Nested Radicals

I know that: $$\sqrt{2+\sqrt{2+\sqrt{2+...\sqrt{2}\; (upto\; n\; times)}}}=2\cos(2^{-n-1}\:\pi)$$ I was wondering whether such a formula exists for $$\sqrt{3+\sqrt{3+\sqrt{3+...\sqrt{3}\; (upto\; n\; times)}}}$$ or in general…
Arya Das
  • 123
7
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If $x^2+4y^2=4.$ Then find range of $ x^2+y^2-xy$

If $x^2+4y^2=4.$ Then find range of $ x^2+y^2-xy$ $\bf{My\; Try::}$ Given $$x^2+4y^2 = 4\Rightarrow \frac{x^2}{4}+\frac{y^2}{1} = 1$$, So parametric Coordinate for Ellipse are $x = 2\cos \phi$ and $y = \sin \phi$. Now Let $$f\left(x,y\right) =…
juantheron
  • 53,015
7
votes
2 answers

Can $f(g(x)) = x$ if $g(f(x))$ is not equal to $x$?

Is it possible that there are functions $f(x)$ and $g(x)$ where $f(g(x)) = x$ and $g(f(x))$ does not equal $x$? If so, is there a pattern or general rule for them? If not, what is the proof that there is not? This originated when a math teacher said…
Matthew Torrence
  • 73
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1 answer

Set of Possible Polynomials

$$x^6\pm x^5\pm x^4 \pm x^3 \pm x^2 \pm x \pm 1 = p(x)$$ What is the set $A$ of possible polynomials from the class of polynomials $p(x)$ such that the polynomial only has real roots. I am confused over how to approach the problem. Should I use…
Phys_asr
  • 101
7
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2 answers

real values of $x$ in $\sqrt{5-x} = 5-x^2$.

Calculate the real solutions $x\in\mathbb{R}$ to $$ \sqrt{5-x} = 5-x^2 $$ My Attempt: We know that $5-x\geq 0$ and thus $x\leq 5$ and $$ \begin{align} 5-x^2&\geq 0\\ x^2-\left(\sqrt{5}\right)^2&\leq 0 \end{align} $$ which implies that…
juantheron
  • 53,015
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