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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Solving system $\{x+xy+y=223,~x^2 y+x y^2=5460\}$

$$x+xy+y=223$$ $$x^2 y+x y^2=5460$$ I need to find the integer solutions to this equation. However, from the looks of it a simple substitution and solve will be difficult, so it seems that clever manipulations might be necessary. I noticed that…
1110101001
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Question on Algebra

Let $P(x)$ be a fourth degree polynomial with coefficient of leading term be $1$ and $P(1) = P(2) = P(3) = 0$, then find the value of $P(0) + P(4)$ .
Achal Kumar
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The intuition behind slant asymptotes

I'm asking this question to get a better understanding of oblique asymptotes. As regards vertical asymptotes, I know that they represent the numbers that make a function undefined. An example of this is a $0$ in the denominator, or a negative…
Andreas Grech
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Why do exponents not distribute over addition?

I understand that exponents don't distribute over addition and have seen plenty of examples i.e. $$ (x + y)^2\neq x^2 + y^2 $$ but I'm wondering why that is. Multiplication distributes over addition e.g. $3(2+3) = 3(2) + 3(3)$ so if an exponent is…
Milo
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How to solve $(1-\tan^2x)\sec^2x+2^{\tan^2x}=0$

I am stuck the following problem : Solve the following problem: $(1-\tan^2x)\sec^2x+2^{\tan^2x}=0$ Taking $p=\tan^2x,$ we get $(1-p^2)+2^p=0$. Now,I do not know how to go from here. Simple observation says $x=2n\pi \pm \frac{\pi}{3}$,satisfies the…
learner
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determining the maximum of a rational function

I was going over some practice problems and got stuck with this one: I am supposed to find the maximum of the function: $$\dfrac{x}{x^2+1}$$ on the interval $(0,4)$. All I can think of is to try and plug in values from the given interval . Is…
Adam
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Coefficients of a Rational Function

This is essentially a definition question. Given a rational function $\frac{p(x)}{q(x)}$, what would the $x^k$ coefficient of this rational function mean (in particular for the negative $k$'s). Is there some sort of expansion $$\frac{p(x)}{q(x)} =…
user898033
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Derivation of slope of line formula

The formula for slope of a line as we know: $y_2 - y_1/x_2 - x_1$ or just rise / run What is the derivation for this formula? E.g. Why is it not rise times run for example?
cpx
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Eliminate $x,y,z$ between the equations.

Eliminate $x,y,z$ between the equations $$\dfrac{y}{z}-\dfrac{z}{y}=a,\dfrac{z}{x}-\dfrac{x}{z}=b,\dfrac{x}{y}-\dfrac{y}{x}=c$$. I understand that if I can somehow find the values of $\dfrac{x}{y},\dfrac{y}{z},\dfrac{z}{x}$ in terms of $a,b,c$…
Hawk
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Find the Solution Set to the Exponential Equation

I have the following exponential equation: $$3^{2x}+3^{x}-2=0$$ I use the product rule to separate the exponent from the base and that resulted in: $$2x \cdot \ln(3) + x \cdot \ln(3) = 2$$ Then I factored out the $x$ from the two terms and divided…
ProfessorStealth
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Interval Notation and Infinity- Closed or Open?

I was wondering whether I should use closed $[-\infty, \infty ]$ or open $(-\infty, \infty )$ notation when representing the infinity sign in interval notation. My teacher says to use open symbols because infinity has no end, but I was also taught…
Salads
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Is $x = -x$ a contradiction?

So I was doing this problem. And I got to what I thought was a weird result. can anyone explain to me why this makes sense? $$\begin{align} 5e^x &= 5e^{-x} \\ e^x &= e^{-x} \\ \ln (e^x) &= \ln (e^{-x}) \\ x &= -x \end{align}$$ This last step is…
nachos
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If $a+b+c = 7\;\;,a^2+b^2+c^2 = 23$, then $a^3+b^3+c^3=$

If $a,b,c\in \mathbb{R}$ and $a+b+c = 7\;\;,a^2+b^2+c^2 = 23$ and $\displaystyle \frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1} = 31$. Then $a^3+b^3+c^3 = $ $\bf{My\; Trial\; Solution::}$ Given $a^2+b^2+c^2 = 23$ and $a+b+c = 7\Rightarrow (a+b+c)^2 =…
juantheron
  • 53,015
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Solving for $x$ in $a^x / x = b$

I've ended up with an equation of the form $a^x / x = b$ and I'm trying to solve for $x$ but I can't isolate it. I always end up with one of the $x$'s as the exponent of $e$ or in a log function.
jtht
  • 217
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$(3^x-2^x)^2+3^x-2^x\leq 5^x$

Solve the following inequality $$(3^x-2^x)^2+3^x-2^x\leq 5^x$$ I'm having difficulty in solving this inequality because there are multiple exponential with different basis. I thought, however, the replacement: $$y=3^x-2^x$$ and so: $$y^2+y\leq…
Mark
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