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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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How long would a track be?

A and B are swimming in lanes right next to each other, but in different directions. They both start at the same time, and they pass each other after person A has swam 84 feet. When they reach the end, they turn around and swim back, meeting again…
Eric Lee
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The sum of $2012$ positive terms is $10.$

A sequence has $2012$ positive terms adding to $10$ Prove that the sum of product of two consecutive terms is less than $25.$ I've tried using the Cauchy-Schwarz inequality but couldn't find the proof. Also find the maximum value (where equality…
user481779
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How to calculate the opposite of a number?

So in general we find the opposite of a number by adding a '-' sign. We're able to add the '-' sign to negative numbers as well, but I was wondering if this is correct: Say we'd want to find the opposite of -7; $$-(-7)$$ Is it true that one finds…
yokihadu
  • 321
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Order of operations in evaluating a polynomial

I have the following function $$f(x) = 3x^3 - 5x^2 - 4x + 4.$$ I would like to find the value of $f(x)$ when $x = -3$. I have ordered the equation in the following way. $$f(-3) = (3 \times -3^3) - (5 \times -3^2) - (4 \times -3) + 4 = -28.$$ Can…
cocacola09
  • 193
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Finding the exact value of a radical

How do I show that $\sqrt{97 +56\sqrt3}$ reduces to $7 +4\sqrt3?$. Without knowing intitially that it reduces to that value.
user3753
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Simple Algebraic Manipulation I am Stuck On?

For some reason I can't seem to see why $$\frac{az+b}{cz+d} = \frac{bc-ad}{c^2\left(z+\frac{d}{c}\right)} + \frac{a}{c}.$$ Its been so many years since I've taken Algebra I, so when this was stated in class, without even the motions of manipulation,…
I Love Cake
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Any simplification of $\log_e{(1+e^x)}?$

Is there any simplification or other interesting transformation of: $$\log_e{(1+e^x)}$$ (where $x \in \mathbb{R}$) ?
Andrew Tomazos
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Does $x^{-1}$ have -1 zeroes?

I asked How $y=x^4+1$ could have 4 zeroes yesterday, and figured it out on my own pretty quickly. I did some more thinking since then, and realized that according to the fundamental theorem of algebra, $0=x^{-1}$ has -1 zeroes (solutions?), because…
Travis
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Completing the square for quadratic equation

I'm learning algebra (at age $66$) with home schooling materials. I am doing well and enjoying it. My question is this: The material says I can use "Completing the square" to solve any quadratic equation. I solved this equation $$6x^2+24x=0$$ by…
James Ward
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Find $P(n+1)$ where $P(x)$ is a $n$ degree polynomial.

A polynomial $P(x)$ of $n$ degree satisfies $P(k)=2^k$ for $k = 0,1,2,3......,n$. Find the value of $P(n+1)$. How can I proceed in solving such problems.
Blaise Thunderstorm
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Show that the equation $\sqrt{ax+\alpha}+\sqrt{bx+\beta}+\sqrt{cx+\gamma}=0$ reduces to a simple equation.

I have got a question which looks like follow: Show that the equation $\sqrt{ax+\alpha}+\sqrt{bx+\beta}+\sqrt{cx+\gamma}=0$ reduces to a simple equation if $\sqrt{a}\pm\sqrt{b}\pm\sqrt{c}=0$. I am totally confused and don't even know from where…
Vidyanshu Mishra
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$\sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}} = \frac{1+\sqrt{5}}{2} = \phi$, is this a coincidence?

I was playing around with square roots today when I "discovered" this. $\sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}} = x$ $\sqrt{1 + x} = x$ $1 + x = x^2$ Which, via the quadratic formula, leads me to the golden ratio. Is there any significance to this…
Alshazgir
  • 185
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infinity sum of numbers

If we have a series of numbers $$1^5 + 2^5 + 3^5 + \cdots + (10^n)^5.$$ Final sum of the series is approximately equal $16666\ldots$ . If there is more and more numbers in the series is the result of closer and closer to $16666\ldots$ . For example…
marko
  • 79
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Difference between $((a+b)/2)^2$ and $ab$?

I was thinking about the difference between the area of a rectangle that is not a square, and a square with sides whose lengths are at the midpoint between the lengths of $a$ and $b$. I did some algebraic manipulation and it seems that the…
Hockeyfan19
  • 643
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Solve $x^2+x^2t^2+2xt^2+t^2=1$ for $x$.

Solving $$x^2+x^2t^2+2xt^2+t^2=1$$ for $x$ yields $$x=\frac{1-t^2}{t^2+1}.$$ I had to use software because I could not manipulate it algebraically, and sadly the program did not tell me how it reached that solution. What is the method?
Susanne Rynders
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