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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.

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Square with variable $x$ inside

I am learning about how to calculate the length of a path with integration. The equation is: $$\sqrt{1+\Big (\frac{dy}{dx} \Big)^2} $$ So I have to integrate it between $a$ and $b$. In my book I have an example, I understand it but don't know how…
Andres
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Value of $a^2 + b^2$ given $a^3 - 3ab^2 = 44$ and $b^3 - 3a^2 b = 8$

If $a$ and $b$ are real numbers such that $a^3-3ab^2=44$ and $b^3-3a^2b=8$ what is value of $a^2+b^2$? I have tried by adding and subtracting these equations, but can't find anything.
chaos
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How is substituting $-y$ for $y$ with $y \neq 0$ legal?

I'm currently going through Spivak's Calculus, and I got across the following problem: Prove that $x^3+y^3=(x+y)(x^2-xy+y^2)$. To prove it, the author wants to use another problem that was proved earlier, i.e $x^3-y^3 = (x-y)(x^2+xy+y^2)$ The…
Aayush Agrawal
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Find the vertical and horizontal asymptotes of the function.

I am asked to find the vertical and horizontal asymptotes of the equation: $$f(x)=(a^{-1}+x^{-1})^{-1}$$ I simplify this to $$f(x)=\frac{1}{a^{-1}+x^{-1}}$$ $$f(x)=a^1+x^1$$$$f(x)=a+x$$Which is some constant, graphed as horizontal line - that will…
Kurt
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Why isn't $\frac{0^{0!}}{0!^0}$ not undefined?

I got into a big argument with my teacher about this. I am saying that it is undefined because every time I work it out, I end up getting $\frac{0}{0}$ which I know to be undefined.
6thYearWizard
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How can I calculate a non-integer power of a number?

Integer powers are easy to calculate by repeated application of multiplication. However if a power is not an integer then I always need to use my calculator. How can I calculate a non-integer power without a calculator? For example, how does one…
user296861
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Functional equation $g(n)=f(f(n))+1$

Two functions $f,g:\mathbb{N} \to \mathbb{N}$ meet the following conditions. For $A=\{f(n)|n \in \mathbb{N}\}, B=\{g(n)|n \in \mathbb{N}\}$, $A \cap B=\emptyset$ $A \cup B= \mathbb{N}$ $g(n)=f(f(n))+1 \quad \forall n \in \mathbb{N}$ $f(n+1)>f(n)$,…
user295651
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Solve the general cubic by factoring

After learning Cardano's solution of the cubic, I decided to look at Ferrari's solution of the quartic (both articles on Wikipedia). At the end of the article on the quartic function was an an alternate method to solve the cubic by factoring into…
hedgepig
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$x \propto y^2$ Vs $x \propto y$

$$x \propto y^2$$ How is it different from saying: $$x \propto y$$ That is; when we say that Two variables are proportional then it means that two variables are related such that when one is zero other is too. And change in one variable is…
Sufyan Naeem
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values of $a$ for which function $f(x)$ is periodic function.

The number of real values of $a$ for which $f(x) = x^a+\sin x-ax$ is a periodic function. $\bf{My\; Try::}$ If function $f(x)$ is Periodic function , Then it must satisfy the condition $f(x+T) = f(x)\;,$ Where $T$ is period of that function…
juantheron
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Solving $\sqrt{x^2 +2x + 1}-\sqrt{x^2-4x+4}=3$

My question is: Solve $\sqrt{x^2 +2x + 1}-\sqrt{x^2-4x+4}=3$ I deduced that:$LHS= x+1-(x-2)$ I am unable to solve this equation. I would like to get some hints to solve it.
mgh
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How to find $A_1A_2 + \cdots + A_{2010} A_{2011}$, where $A_{n+1} = \frac{1}{1+\frac{1}{A_n}}$

My question is: If $$A_{n+1} = \frac{1}{1+\frac{1}{A_n}}$$ ($n\in\mathbb{N}$) and $A_1=1$, then find the value of: $$A_1A_2 + A_2A_3 + A_3A_4 + \cdots + A_{2010} A_{2011}.$$ Please I would like to get some hints to solve this question.
mgh
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Solving $\sqrt[3]{x^2} + \sqrt[3]{x} = 2$

My sister asked me for some help on her algebra homework the other day, and I was stumped by her question. The problem is to find the root of $\sqrt[3]{x^2} + \sqrt[3]{x} = 2$. The internet tells me that x is 1, but I can't seem to figure out…
munk
  • 215
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Is there a way to solve $\sqrt a + \sqrt b = \sqrt n$ analytically?

Often in (high school) math competitions, there are equations that look very simple, yet are non-trivial to solve. One such problem is $$ \sqrt a + \sqrt b = \sqrt n $$ where $a, b \in \mathbb{Z}$ and $n$ is usually an arbitrary value, like a year…
user288158
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Extraneous Roots

When dealing with extraneous roots and checking them, if you find that one of the, two roots let's say, is extraneous, then does this imply that the other root is real or would one need to check the other root as well...so potentially it has no…
Liam Cooney
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