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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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Two proofs involving Harmonic Mean

If H is the harmonic mean between $a$ and $b$,then show that $$\frac{1}{H-a}+\frac{1}{H-b} = \frac{1}{a} + \frac{1}{b}$$ and $$\frac{H+a}{H-a}+\frac{H+b}{H-b} = 2$$ I substituted $\displaystyle H = \frac{2ab}{a+b}$, then tried some algebraic…
Max
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For what values of $a$ does $1/(1+1/x) = a$ have no solution for $x$? Similarly, $ (6x-a)/(x-3) = 3$?

For what values of $a$ does this equation have no solution in $x$? $$\frac1{\;1+\dfrac1x\;} = a$$ Similarly, for what value of $a$ does this equation have no solution in $x$? $$\frac{6x-a}{x-3} = 3$$ I get only the value $a = 1$ for the first,…
Sami A
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Sum of resistors in parallel is smaller than resistance of smallest resistor

The motivation of this question is that when you connect two or more resistors in parallel, then the total resistance is smaller than each of the resistors. So in general, I guess I want to prove that given a set of numbers $\{a, b, c, \cdots,…
PhysicsMathsLove
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Quadratic in one

Let $a+b+c=0$. Does $1^2a+1b+c=0 \implies b^2 \geq 4ac \ $ ? Obviously we know the inequality holds in general, but does considering a "quadratic in one" imply this?
wandering
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Extraneous solution from substituting in equations

I came across this example of how you could end up with an extraneous solution but I was wondering how it arose. We have the equation: $$x^2+x+1=0 $$ Since x=0 does not satisfy the equation, you can divide by x on both sides which yields:…
user761982
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How to factorize $a^2-2ab+a^2b-2b^2$?

I have been stuck on factorizing this: $$a^2-2ab+a^2b-2b^2$$ I thought I could solve it by making $(a+b)$ as one factor but it didn't work then I tried to add and deduct some terms which that didn't lead me to anything either. I don't really know…
Peter
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Graphing inequalities

$xy\geq 4$ The answer my book gives me is a hyperbola, but I always thought a hyperbola was a difference between two variables, how does this work?
Adam
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What's the difference between these two solutions?

$4x^2 = 4x$ If we divide both sides by $4x$ we get $x = 1$ but if we do this: $4x^2 - 4x = 0$ $4x(x-1) = 0$ we get $x = 1, x = 0$ I guess the latter solution is the correct one, but I want to understand why the first one is wrong, and since…
Guest
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Simplifying simple radicals $\sqrt{\frac{1}{a}}$

I'm having a problems simplifying this apparently simple radical: $\sqrt{\frac{1}{a}}$ The book I'm working through gives the answer as: $\frac{1}{a}\sqrt{a}$ Could someone break down the steps used to get there? I've managed to answer all the…
DaveG
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Sum of all numbers x such that $(3x^2+9x-2012)^{(x^3-2012x^2-10x+1)} = 1$?

What is the sum of all $x$ such that $(3x^2 + 9x - 2012)^{(x^3-2012x^2-10x+1)} = 1$?
user68472
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Solve $\lceil x\rceil^2+\lceil x+1\rceil^2=25$

How can we find the solution set of $\lceil x\rceil^2+\lceil x+1\rceil^2=25$ where $\lceil x\rceil$ is the ceiling function?
Sachin
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Solve Exponential Equation with square

Solve equality with square $2^{2x}=7\cdot 2^{x+\sqrt{x-1}}+8\cdot 2^{2\sqrt{x-1}}$ $x-1\ge0 \\\sqrt{x-1}=t\ge0\Rightarrow x-1=t^2\Rightarrow x=t^2+1\\2^{2(t^2+1)}=7\cdot2^{t^2+1+t}+2^{3+2t}$ It looks very complicated and I don't know how to move…
vmahth1
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Compound interest formula with regular deposits, solve for time

I cant seem solve the following problem due to my bad memory of logarithms if someone opens an account with initial balance of $B$ interest rate of $R$ and does a constant monthly deposit of $D$ how long before the account balance reaches $X$ ?
user9913
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Given $a+b+c=0$, simplify the following.

I am here again to ask a question about an exercise I saw around but i'm having a lot of trouble with. I know the answer is 3abc, but as in many of my questions, I am interested in the why and how. Given $a+b+c=0$,…
chubakueno
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There exists $2n-1$ pairs $(a_i, a_j)$ s.t. $a_i+a_j\geq 0$.

Let $a_1, a_2,..., a_{2n}\in \mathbb {R} $ s.t. $a_1+a_2+...+a_{2n}=0$. Show that there exists $2n-1$ pairs $(a_i, a_j)$ s.t. $a_i+a_j\geq 0$. My idea: I consider that $a_1,..., a_k\geq 0$ and $a_{k+1},..., a_{2n}<0$(I put them the sign…
Problemsolving
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