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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 squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself?

If squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself? For example the square of $2$ is $2^2=2 \cdot 2=4 $ . But square root of $2$ is not $\frac{2}{2}=1$ .
bluebellae
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Why can't you square both sides of an equation?

Why can't you square both sides of an equation? I've been asked this many times and can never quite give a good, clear, concise answer (for beginning algebra students) in plain language. I just searched the web and still couldn't find a…
Jeff
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16 answers

Intuition for why the difference between $\frac{2x^2-x}{x^2-x+1}$ and $\frac{x-2}{x^2-x+1}$ is a constant?

Why is the difference between these two functions a constant? $$f(x)=\frac{2x^2-x}{x^2-x+1}$$ $$g(x)=\frac{x-2}{x^2-x+1}$$ Since the denominators are equal and the numerators differ in degree I would never have thought the difference of these…
GambitSquared
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Algebra: What allows us to do the same thing to both sides of an equation?

I understand that the expressions on both sides of an equal sign are the same entity, and I know that when you modify one side, the other must be changed because it is referring to the same thing. What I do not understand is why making a new…
aaax2178
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3 answers

If there are $74$ heads and $196$ legs, how many horses and humans are there?

I was going through some problems then I arrived at this question which I couldn't solve. Does anyone know the answer to this question? One day, a person went to a horse racing area. Instead of counting the number of humans and horses, he counted…
Prateek
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5 answers

Why should you never divide both sides by a variable when solving an equation?

I'm currently working through an algebra book, and during the chapter about rational expressions and inequalities, the author has a side note in which he states: Never divide both sides of the equation by a variable, even if you're doing it to…
Andreas Grech
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38
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2 answers

Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $?

Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $? I'm currently studying indices/exponents, and this is a law that I was told to accept without much proof or explanation, could someone explain the reasoning behind this. Thank you.
seeker
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A pencil approach to find $\sum \limits_{i=1}^{69} \sqrt{\left( 1+\frac{1}{i^2}+\frac{1}{(i+1)^2}\right)}$

What is the fastest, paper-pencil method of finding $$\sum \limits_{i=1}^{69} \sqrt{\left( 1+\frac{1}{i^2}+\frac{1}{(i+1)^2}\right)}?$$ This is actually a quantitative aptitude problem, and hence the solutions should be fast enough and probably…
Quixotic
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Why does cancelling change this equation?

The equation $$2t^2 + t^3 = t^4$$ is satisfied by $t = 0$ But if you cancel a $t^2$ on both sides, making it $$2 + t = t^2$$ $t = 0$ is no longer a solution. What gives? I thought nothing really changed, so the same solutions should apply. Thanks
Dyldo42
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6 answers

Explanation of method for showing that $\frac{0}{0}$ is undefined

(This was asked due to the comments and downvotes on this Stackoverflow answer. I am not that good at maths, so was wondering if I had made any basic mistakes) Ignoring limits, I would like to know if this is a valid explanation for why $\frac00$ is…
Jacob
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34
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9 answers

Proving $x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$

In Spivak's Calculus 3rd Edition, there is an exercise to prove the following: $$x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$$ I can't seem to get the answer. Either I've gone wrong somewhere, I'm overlooking something, or…
jamesbrewr
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7 answers

Why is $\frac{1}{\frac{1}{0}}$ undefined?

Is the fraction $$\frac{1}{\frac{1}{0}}$$ undefined? I know that division by zero is usually prohibited, but since dividing a number by a fraction yields the same result as multiplying the number by the fraction's reciprocal, you could argue…
Peter Olson
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3 answers

How to calculate percentage of value inside arbitrary range?

So pardon if this is a simple question... I have a slider that returns a value in a given range, so: min: 174 max: 424 slider current value: 230 I want to treat my min value as 0% and my max value as 100%. What formula can I use to calculate the…
neezer
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28
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11 answers

Can you cancel out a term if equal to zero?

quick question here: In my proofs class we had a problem that after a little work we end up with: $x(x-y)=(x+y)(x-y)$ where $ x = y $. Now, I know this is pretty basic, but my teacher said that for the next step, one cannot cancel out $(x-y)$ from…
Descoladan
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28
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3 answers

How to calculate the percentage of increase/decrease with negative numbers?

I feel like an idiot for asking this but i can't get my formula to work with negative numbers assume you want to know the percentage of an increase/decrease between numbers 2.39 1.79 =100-(1.79/2.39*100)=> which is 25.1% decrease but…
master of puppets
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