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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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Find all integers $m$ such that $\frac{1}{m}=\frac{1}{\lfloor 2x \rfloor}+\frac{1}{\lfloor 5x \rfloor} $

How would you determine all integers $m$ such that the following is true? $$\frac{1}{m}=\frac{1}{\lfloor 2x \rfloor}+\frac{1}{\lfloor 5x \rfloor} .$$ Note that $\lfloor \cdot \rfloor$ means the greatest integer function. Also, $x$ must be a…
Evan Warner
  • 31
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Why variables in directly proportinality are multipiled

Why variables (RHS) in directly proportionality are always multiplied. Suppose the Newton's 2nd law $$F \propto m$$ $$F \propto a$$ $$F \propto m*a$$ Please don't give a rigorous proof. I just want to understand it intuitively.
Atinesh
  • 1,239
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If the mean of $x,y,z$ is $3x$, what is the mean of $y$ and $z$?

$x, y, z$ are three numbers. The mean of $x, y, z$ is $3x$. What is the mean of $y$ and $z$? Is the answer $4x$?
Patrick Beardmore
  • 478
3
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3 answers

How do I solve $(x^3-4x^2+5x-6)/(x^2-x-6)=4$ algebraically?

How do I solve $\frac {(x^3-4x^2+5x-6)}{(x^2-x-6)=4}$ algebraically? I tried: $4(x^2-x-6)=x^3-4x^2+5x-6$ $4x^2-4x-24=x^3-4x^2+5x-6$ $x^3-8x^2+9x+18=0$ I don't know how to solve this algebraically.
Thas
  • 41
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5 answers

Why is this system of equations unsolvable?

The system is: \begin{cases} 3a + 5b = 2\\ 15a + 15b = ? \end{cases} Can't we just do: \begin{gather} 3a = 2 - 5b\\[2ex] a = \frac{2 - 5b}{3} \end{gather} Then we plug in $a$ in terms of $b$ into the second equation, which…
jeremy radcliff
  • 4,805
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Solving $(\cos\alpha-\lambda)^2+\sin^2\alpha=0$ for $\lambda$.

I am trying to solve $$(\cos\alpha-\lambda)^2+\sin^2\alpha=0$$ for $\lambda$. Expanding and using the identity $\sin^2x+\cos^2x=1$ yields $$\lambda^2-2\lambda\cos\alpha+1 = 0$$ and using the quadratic formula gives…
Peter Olson
  • 2,253
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How to expand binomials with odd numbered exponents?

I know how to expand when the expression is like: $$a^2-b\qquad\text{ which expands to }\qquad(a+b^{1/2})(a-b^{1/2})$$ or like $$a^4-b\qquad\text{ which expands to }\qquad(a^2+b^{1/2})(a^2-b^{1/2})$$ But what is the method / formula to expand…
Anon
  • 33
3
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3 answers

Reason what 2^k in a sigma will do

I am trying to solve the following calculation without a calculator: $$\sum_{k=0}^82^k{8\choose k}$$ The first part: $$\sum_{k=0}^8{8\choose k}$$ is equal to $2^8$. I already know that the answer will be $3^8$. How did the $2^k$ transform the…
Dagob
  • 133
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Need help with a simple word problem

This is probably a dumb question but I'm stuck. So here goes: The dimensions of a plot of land are in a 3:4 ration. If the area of the plot of land measures 1072 m^2 and is also surrounded by a uniform walkway that has a width of 2m. Find the…
Argus
  • 369
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votes
2 answers

Find the function given the equation $(g \circ f)(x) = x + 5$

Let $g(x) = 4x + 1$. Find $f(x)$, given that the equation $(g \circ f)(x) = x + 5$ is true for all values of $x$. From my text, it looks like this would be expressed as $g[f(x)]$, but I don't see how you can determine the definition of $f(x)$. Do I…
erimar77
  • 741
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3 answers

Need help to solve taylor series of $e^{\sin x}$

How to derive the taylor series of $e^{\sin x}$, up to $x^5$? i just don't know how to get the answer $$f(x) = 1 + x + \frac{x^2}{2} - \frac{x^4}{8} -\frac{x^5}{15}$$ really need some help. Thanks
user139519
  • 31
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votes
4 answers

Why is $\log_{49}\sqrt{7}=\frac{1}{4}?$

I cannot understand why $\log_{49}(\sqrt{ 7})= \frac{1}{4}$. If I take the $4$th root of $49$, I don't get $7$. What I am not comprehending?
Frustrated with Math
  • 131
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2 answers

Complete the square of a 4th degree expression $k^4 + 2k^3 -k^2 - 2k + 1$

I was reading the solution of a problem and I can't figure out how to go from here: $k^4 + 2k^3 -k^2 - 2k + 1$ to: $((k^2 + k) -1)^2$ I guess it's using the complete-the-square method but I've only used that with quadratic expressions.
mathnoob
  • 31
3
votes
3 answers

Factoring $r^8-1$

I am trying to factor $r^8-1$ after having just learned about difference and sum of cubes. I don't know what to do next though as this doesn't seem like a difference of cubes. How do I factor this in the intended way? I suppose I could just do brute…
nitrous2
  • 2,451
3
votes
1 answer

Is the suggested answer in the book wrong? (sign diagram)

The function is as follows $-2x^3+6x-4$ The sign diagram for that in the book is as follows: As seen on the pic and graph it is only positive before -2, NOT after 1. Or did I miss something?
Algific
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