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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
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Show that $n^4-20n^2+4$ is composite when $n$ is any integer.

Crux Mathematicorum June 1978, Show that $n^4-20n^2+4$ is composite when $n$ is any integer. My question: What is meant by 'composite'? Besides, I found out that this expression could be factorised into $(n^2-4n-2)(n^2+4n-2)$. Is this related…
Mathxx
  • 7,570
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Solving 4 Quadratic Simultaneous Equations

Consider the equations: $$\begin{cases} ab+c+d=3\\ bc+d+a=5\\ cd+a+b=2\\ da+c+b=6 \end{cases}$$ Where $a,b,c,d \in \mathbb{R}$. How can we find $a,b,c,d$? The furthest I've got is by adding the first two equations and the last two equations…
preferred_anon
  • 17,121
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Show that $y =e-x^2e^x$ has an $x$-intercept at $x = 1$

Show that $$y =e-x^2e^x$$ has an $x$-intercept at $x = 1$. My Process: $x$-intercept when $y=0$, so \begin{align*} e-x^2e^x&=0\\ -x^2&=0\\ x&=0 \end{align*} However when I graphed this function online, I could see that $x$-intercept is $1$ but I…
Utsav
  • 389
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Solve for $x$ in $\frac{x^2+12x+36}{x^2+4x-12}=2$

$\frac{x^2+12x+36}{x^2+4x-12}=2$ I factored top and bottom to $\frac{(x+6)(x+6)}{(x+6)(x-2)}=2$ and eliminated the common factor (x+6) for $\frac{x+6}{x-2}=2$ then $x+6=2(x-2)$ and $x+6=2x-4$ and $6+4=2x-x$ and finally $x = 10$ Plugging this…
user3255759
  • 53
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How many three digit numbers have all three digits that are different?

How do I solve this? Please explain in detail. I am told the answer is $9\times9\times8$ but I do not understand how that would lead to the solution.
Billyyyyy
  • 59
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Solve the equation ${\sqrt{4x^2 + 5x+1}} - 2{\sqrt {x^2-x+1}} = 9x-3$

I tried factoring the expression inside the square root, but that does not seem to help. Squaring the equation makes it even more terrible. Can anyone provide a hint about what should be done?
user399078
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4 answers

Prove that $\frac{bc}{a(a-b)(a-c)}+\frac{ca}{b(b-c)(b-a)}+\frac{ab}{c(c-a)(c-b)}=\frac{ab+bc+ca}{abc}$

Prove that $$\frac{bc}{a(a-b)(a-c)}+\frac{ca}{b(b-c)(b-a)}+\frac{ab}{c(c-a)(c-b)}=\frac{ab+bc+ca}{abc}$$ To do this I unified three terms on the left side with a common denominator and then factored the numerator (with the aid of Wolfram Alpha...…
Kay K.
  • 9,931
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4 answers

Find $x^2+y^2+z^2$ if $x+y+z=0$ and $\sin{x}+\sin{y}+\sin{z}=0$ and $\cos{x}+\cos{y}+\cos{z}=0$ for any $x,y,z \in [-\pi,\pi]$

Find $x^2+y^2+z^2$ if $x+y+z=0$ and $\sin{x}+\sin{y}+\sin{z}=0$ and $\cos{x}+\cos{y}+\cos{z}=0$ for any $x,y,z \in [-\pi,\pi]$. My attempt:I found one answer $x=0,y=\frac{2\pi}{3},z=-\frac{2\pi}{3}$ which gives the answer…
Taha Akbari
  • 3,559
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How to solve this equation with another way?

I have the equation $$\left(\dfrac{x+1}{x-2}\right)^2 + \left(\dfrac{x+1}{x-3}\right) = 12\left(\dfrac{x-2}{x-3}\right)^2$$ I want to solve this equation in the set of all real numbers. First way. Put $t = \dfrac{x-2}{x-3}$, and then $x =…
minthao_2011
  • 1,885
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2 answers

Solve $\left(\sqrt{\sqrt{x^2-5x+8}+\sqrt{x^2-5x+6}} \right)^x + \left(\sqrt{\sqrt{x^2-5x+8}-\sqrt{x^2-5x+6}} \right)^x = 2^{\frac{x+4}{4}} $

Solve $$\left(\sqrt{\sqrt{x^2-5x+8}+\sqrt{x^2-5x+6}} \right)^x + \left(\sqrt{\sqrt{x^2-5x+8}-\sqrt{x^2-5x+6}} \right)^x = 2^{\frac{x+4}{4}} $$ Preface; I think there should be an algebraic method of solving this equation for $x$ since graphing…
bigfocalchord
  • 5,597
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Solving two equations of 2nd degree and 2 variables

I spent hours trying to solve this system of equations, they're two equations with two variables, and both are 2nd degree equations, I don't know, both equal zero, so each one equals the other, but I came to a step where I'm stuck. The system…
Stephen Alexander
  • 161
5
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3 answers

Solution(s) to 'power equations'

I'm not sure a 'power equation' is the right name for the equation I'd like to know more about (and, specifically, about its solutions), but I don't know the 'proper' way to name it. The simplest 'power polynomial' is : $P(x) = x^x $. The simplest…
Max Muller
  • 7,006
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4 answers

Prove $4a^2+b^2+1\ge2ab+2a+b$

Prove $4a^2+b^2+1\ge2ab+2a+b$ $4a^2+b^2+1-2ab-2a-b\ge0$ $(2)^2(a)^2+(b)^2+1-2ab-2a-b\ge0$ Any help from here? I am not seeing how this can be factored
Michelle Stasso
  • 193
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How to calculate the percentage of a value contemplating negative numbers?

min = 0 max = 100 v = 50 Given those values, I can calculate the percentage of "completion" for v within the range [min,max] like this: (100.0 / max) * v And I get 50% - the desired value. As you can see, I don't even use the min variable. So I…
Saturn
  • 7,191
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Help with a formula for Google Adwords

I am creating a spreadsheet to calculate Adword formulas and I am stuck in how to calculate the Monthly Net Income for each keyword. I have created a formula which calculates it but can't figure out how to limit the Monthly Budget. The formula I've…
Xavi Esteve
  • 153
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