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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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Find $f(x)$ from $f(3x + 1)$

The problem that I have to solve is: If the following function is valid for every value of $x$ $$f(3x + 1) = 9x^2 + 3x$$ find the function $f(x)$ and prove that for every $x\in\mathbb R$ the following is valid: $$f(2x) - 4f(x) = 2x$$
Chris
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Proving an algebraic identity

Prove: $$(a + b + c)(ab + bc + ca) - abc = (a + b)(b + c)(c + a)$$ Problem: I am not sure how to proceed after expanding the brackets on the RHS. I am not sure if I also expanded correctly. My solution is:
IgorTheGr8M8
  • 71
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3 answers

Factorization of $x^5+x$

I need to make the decomposition in $\mathbb{R}$ of: $$x^5+x$$ Here my steps: $$x(x^4+1)$$ $$x\big[ (x^2)^2+1 \big]$$ $$x\big[(x^2+1)^2-2x^2\big]$$ how should I proceed?
Christian Giupponi
  • 687
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4 answers

Real solution of the equation $\sqrt{a+\sqrt{a-x}} = x\;,$ If $a>0$

For a real number $a>0\;,$ How many real solution of the equation $\sqrt{a+\sqrt{a-x}} = x$ $\bf{My\; Try::}$ We can Write $\sqrt{a+\sqrt{a-x}} = x$ as $a+\sqrt{a-x}=x^2$ So we get $(x^2-a)=\sqrt{a-x}\Rightarrow (x^2-a)^2 = a-x\;,$ Where $x
juantheron
  • 53,015
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Find the value of $ab+ 2cb+\sqrt3 ac$?

Three positive real numbers $a,b,c$ satisfy the equations $a^2+\sqrt3 ab+b^2=25$, $b^2+c^2=9$ and $a^2+ac+c^2=16$ .Then find the value of $ab+ 2cb+\sqrt3 ac$? Is there some way to find the desired value without actually finding values of $a,b,c$…
H.P. Das
  • 805
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Proportional to 2 Separate Variables vs. Proportional to Product of 2 Variables

I've commonly seen the following in physics and math textbooks, but never understood how it is mathematically deduced: $A \propto B$ $\space$ and $\space$ $A \propto C \space\space\space \implies \space\space\space A \propto BC.$ Could someone walk…
chevestong
  • 325
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Algebraic shortcut to find $a^n + b ^n$?

Recently, I found this problem online: Given $a+b=1$ and $a^2+b^2=2$, find $a^7+b^7$. Although I could've solved it by substituting the first equation into the second and then using the quadratic formula; the way the question was set up, I…
Shuri2060
  • 4,353
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Solving $e^x = 6x$ for $x$ without a graph.

Throughout my high school career I was always told that an equation of this sort ( $e^x = 6x$ for example) couldn't be solved algebraically. However I feel that there may be a way (and you may be out there saying "of course there is a way") I know…
WaveX
  • 5,440
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Decomposition of $x^n-y^n$

As a part of textbook assignment I was asked to prove that $x^2-y^2=(x+y)(x-y)$, and I did so as follows: $$x^2-y^2=x^2-y^2+xy-xy=x(x+y)-y(x+y)=(x+y)(x-y)$$ Later, I used similar method to decompose $x^3-y^3$, with different…
Misha.P
  • 221
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Analytical solution for rational equality, square root in denominator on both sides

So I was trying to solve a problem I saw in a practice set for a 6th-grade math competition, as far as I can remember it. It was a story problem, but I think the solution is the minimum value of $$ \sqrt{x^2 + 25} + \sqrt{(5-x)^2 + 49} $$ for $x$ in…
david
  • 155
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3 answers

Condensation of an expression

There is an expression: $(h+2p)^2(x+2y)^2 - 3p^2(x+2y)^2-3y^2(h+2p)^2+9p^2y^2$ Is there a way to simplify this to the form $l^2 + 4lk + k^2$? I tried to substitute $l = (h+2p)(x+2y)$ and $k= 3py$ and this takes care of the expressions at the two…
user319087
5
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4 answers

Represent $1^5+2^4+3^3+4^2+5^1$ in sum notation

I imagine the sum notation for $$1^5+2^4+3^3+4^2+5^1$$ Would look something like $$ \sum x^y,\ x=1 \text{ to } 5,\ y=5 \text{ to } 1 $$ Is this correct or am I missing something?
Brian
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1 answer

Is my algebra correct here?

so I have the interest equation $$P=C(1+\frac rn)^{nt}$$ and I need to solve for t. So what I did was divide to get rid of C first: $$\frac PC=(1+\frac rn)^{nt}$$ then I took the logarithm to get nt on it's own $$\log_{1+\frac rn}\frac PC =…
AJB_1070179
  • 425
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6 answers

Deriving the Formula for Average Speed (Same distance).

Let me start of by specifying the question: A and B are two towns. Kim covers the distance from A to B on a scooter at 17Km/hr and returns to A on a bicycle at 8km/hr.What is his average speed during the whole journey. I solved this problem by…
Rajeshwar
  • 1,629
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Find the value of $(a^3 + b^3 + c^3)/(abc)$ if $a/b + b/c + c/a = 1$.

Find the value of $$\frac{a^3+b^3+c^3}{abc}\qquad\text{ if }\quad \frac ab + \frac bc + \frac ca = 1.$$ I tried using Cauchy's inequality but it was of no help. Please guide me. $a, b, c$ are real.
Bazinga
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