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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.

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0answers
11 views

Define a variable and an equation for the situation - Why Division and not multiplocation equation?

Dwight wants to buy a camera that is on sale for 25% off. The original price of the camera is $300. What is the amount of the discount? Define a variable, write an equation and solve algebraically. ...
1
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1answer
11 views

Why is this a mutiplication equation VS a division equation?

Circle C Farm has 1,500 chickens. They separate the chickens into 6 different areas. How many chickens are in each area? Define a variable and write an equation. Solve the equation. According to my ...
1
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1answer
54 views

Is there an “inverse” of the dot-product?

The dot product of $[a_1,a_2]$ and $[b_1, b_2]$ is $a_1b_1 + a_2b_2$ (and so on for bigger vectors). What I'm wondering is if there's any definition of a function s.t. the "invdot" product of $[a_1,...
-3
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3answers
44 views

calculate the value in question [on hold]

$x\cdot y\cdot z=5$ $x+y+z=12$ calculate $x^2$+$y^2$+$z^2$
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1answer
42 views

Simplifying a solution to $x^2 = 4 + 2\sqrt{2}$

$$x^2 = 4 + 2\sqrt{2}$$ $$x = 2 + \sqrt{2\sqrt{2}}$$ Neglecting that the result can be negative as well, how should I continue? How can I simplify it?
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0answers
18 views

Summation of product of an exponent and a combination

Please check the image for the question. (https://i.stack.imgur.com/mel2Z.jpg) I thought of trying Vandermonde identity but failed to do so. What will be its solution?
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1answer
29 views

How to find the solutions to an equation

So I have this equation: $$d\left(x\right)=\left(-0.5\pi-\arctan\left(\tan\left(\pi x-0.5\pi\right)\right)\right)^2+\left(-0.5\pi-\arctan\left(\tan\left(\pi\frac{n}{x}-0.5\pi\right)\right)\right)^2-2\...
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3answers
29 views

How do I get $-2$ as a solution to $\frac{3x+1}{x+2}<2$?

I'm stuck at this seemingly simple problem again. Solve the difference $$\frac{3x+1}{x+2}<2$$ I try to solve this in the intuitive way: $$\frac{3x+1}{x+2}<2$$ $$=>3x+1<2(x+2)$$ $$=>...
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0answers
22 views

Are percentages scalable?

I have data made up of percentages but no idea how they were calculated. I am using this data to demonstrate Benford’s Law and so far it does follow this law! But one of the properties of sets that ...
2
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3answers
52 views

What is the minimum time needed for Shubrata and Amlan to eat 252 jackfruits?

Once Shubrata and Amlan ate 50 jackfruits in total for 3 hours and 5 hours respectively. Another time they ate 200 jackfruits for 26 hrs and 10 hrs respectively. Now what is the minimum time ...
4
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1answer
55 views

Find $f$ if $f(x)\leq x$ and $f(x+y)\leq f(x)+f(y)$ for all $x,~y\in \mathbb{R}.$

Find the formula of function $f:\mathbb{R}\to \mathbb{R}$ if: $$f(x)\leq x$$ and $$f(x+y)\leq f(x)+f(y)$$ for all $x,~y\in \mathbb{R}.$ Attempt. Identity function $I(x)=x$ satisfies the needed ...
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2answers
31 views

how to decompose into irreducible real polynomials [on hold]

I need help with decomposing following polynom $$(x +1)^{n} + (x -1)^{n}$$
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3answers
51 views

How can I understand how to draw the graph of the function $|x-a|$? [on hold]

How can I draw the graph of this function in related to $a$ ? $$ f(x) = |x-a| $$
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2answers
31 views

Mathematical induction find values of skipping numbers.

I have a question which is finding the value of a skipping numbers mathematical induction problem. The problem is as follows: Prove by mathematical induction, that $$1^3+2^3+3^3+\dots+n^3=\frac{n^2(n+...
3
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1answer
39 views

Why are the axis of an ellipsoid eigenvectors?

Consider an ellipsoid $\{x| x^TAx = 1\}$. Let $A$ be a real symmetric matrix, and consider the eigen-decomposition $A=P\Lambda P^{-1} =P\Lambda P^T$, where the matrix $P$ is orthogonal because $A$ ...
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2answers
45 views

How to solve this problem? I did not understand how to draw it's diagram. please help.

At the midpoint of line segment $AB$ which is $p$ units long, a perpendicular $MR$ is erected with length $q$ units. An arc is described from $R$ with a radius equal to $\frac{1}{2}AB$, meeting $AB$ ...
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1answer
46 views

Soft question - are these two set notations correct?

Do these sets equal the interval notation given? \begin{align} T&=\{t\in \mathbb{R}:t^2\lt{2}\}=(-\sqrt{2},\sqrt{2})\\ S&=\{s\in \mathbb{R}:s^2\leq{2}\}=[-\sqrt{2},\sqrt{2}] \end{align}
0
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1answer
23 views

How to find the number of squares between a range of numbers that are also divisible by $ 12 $?

I know to find all numbers divisible by $ 12 $ between a range of numbers is something like: $$ \frac{x}{12} - \frac{y}{12} + 1, \quad \text{where}\ x>y .$$ And to find the number of squares ...
2
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1answer
55 views

Should we use parentheses or brackets around functions?

Should we use parentheses or brackets around functions? Is there a widely accepted rule/style? For example, which of the following is the preferred way of writing formulas? $(f(x) + g(x)) * h(x)$ ...
1
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1answer
28 views

Define the variable and equation for the situation.

Walmart has a bike on sale for $$150. The sign next to it says that the price is $25 off the original price. What is the original price? Define the variable and equation for the situation. Solve the ...
2
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2answers
105 views

Quiz Question: Solve for $x$: $9^{2x+1} - 28 (3^{x}) +3 = 0$? [on hold]

I know this is probably quite basic, but I've got my school mathematics quiz tomorrow and I've honestly drawn a blank with this question: Find the sum of the roots of the equation: $9^{2x+1}-28(3^x)+...
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0answers
15 views

Differencial equation of Global mean temperature question in terms of LW SW? [on hold]

You want to develop a model that will predict the temporal evolution and the steady state of the Global Mean Earth temperature (To). Make your life simple: only consider first order term (LW and SW) ...
1
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2answers
55 views

zeros of polynomial.

I need to find zeros(including multiple) of $\ x^3+x^2+1$. When I draw graph I notice 1 only elevate the curve. So I factored $\ x^3 + x^2$ and had $$ x^2(x+1)$$$= (x+0)^2(x+1)$ So according to what ...
3
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1answer
61 views

How to parse $2^{1+2^3}$?

I'm fairly comfortable with arithmetic and high school math overall, but I'm having trouble being 100% sure that $2^{\large {1+2^3}}$ should be parsed as 2^(1+(2^3))...
-1
votes
2answers
57 views

Prove $\frac{a^2b^2}{(a-c)(b-c)}+ \frac{b^2c^2}{(b-a)(c-a)} + \frac{c^2a^2}{(c-b)(a-b)} = ab+bc+ca$

Prove $$\frac{a^2b^2}{(a-c)(b-c)}+ \frac{b^2c^2}{(b-a)(c-a)} + \frac{c^2a^2}{(c-b)(a-b)} = ab+bc+ca$$ Can you help me to prove this equality? When I multiple left side, I get more complicated ...
0
votes
1answer
136 views

How to add numbers with same base but unknown exponents?

I'm taking a university math course despite knowing better. Among the straightforward problems I encountered the following: Simplify: $$5^x +5^{x+2}$$ The answer is supposed to be $26*5^x$ How am ...
1
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2answers
33 views

Unique solution to an Algebraic equation

$$f(x_{2}-x_{1})= f(x_{2}-y)f(y-x_{1})$$ I need to find a unique solution to this algebraic equation. Any hints as to how to proceed. The Exponential function works in this case, but I explicitly need ...
0
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1answer
32 views

What algebraic step is taken here?

We have as part of a Laplace transform: $s^2F(s) - s - 1 + \frac{s^2F(s) - s - 1}{s^{\frac{1}{2}}} + F(s) = \frac{1}{s} + \frac{1}{s^2}\\ \implies F(s)(s^2+s^{\frac{3}{2}} + 1) = (\frac{1}{s} + \frac{...
4
votes
2answers
65 views

Different ways to express $\sqrt{a}+\sqrt{b}$

I had been thinking about this for a long time. I can’t express $\sqrt{a}+\sqrt{b}$ in ways that are useful. (I’m not wanting a formula, but just some ways to express this.) I can only think of $\sqrt{...
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0answers
24 views

Question about parametric equations

Let a curve be paramaterized, so x(t)=g(t) and y(t)=h(t) how do I prove that eliminating the parameter yields a relationships which satisfies the points which (g(t),h(t)) traces (and possibly more)?
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3answers
49 views

What are the zeros of $\frac{x^3+5x^2-14x}{x-2}$? [on hold]

What are the zeros of $\frac{x^3+5x^2-14x}{x-2}$? Obviously zero is correct, but what else?
0
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3answers
43 views

What's the general rule for when applying f(x) = x^2 introduces a “extraneous solution” to a equation? [duplicate]

For example consider the equation $x = 3$ $\implies x^2 = 9 $ $\implies x = 3 \;or\; x = -3$ In this case an extraneous solution is introduced i.e. $ x = -3$ Now for this equation $\sqrt x = 3 $...
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1answer
34 views

Objects moving on a plane

Object 1 starting at point (0,0) on the coordinate plane moves to right at 1 unit per second. Another object (object 2) starts at (x,y), moving half the distance to object 1 each second (following a ...
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2answers
37 views

What would be the correct algebraic expression for this question?

Jasmine bought 2 pounds of apples at $$3 per pound and 4 pounds of bananas at $1 per pound. Write an algebraic expression for the cost of Jasmine's purchase. What would be the correct algebraic ...
2
votes
5answers
71 views

Trick to factor: $n^3 + 6n^2 + 9n + 4$

Is there any way to tell if $$n^3 + 6n^2 + 9n + 4$$ has a factored (condensed) form, and, if so, what that factored form is? If the answer is yes to both, may one explain the more general principle?...
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2answers
45 views

Solving a quadratic equation using the “splitting the middle term” method.

Use splitting the middle term method to solve the below equation. Is there a limitation to this method? $$5b^2-16b+4=0$$
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4answers
74 views

Any way to solve $|x-8| = |2-x|-6$ algebraically?

Everything I've tried has given me $x = 2$ (which is obviously incorrect, since $-6 \neq 6$). The actual answer is $x \geq 8$ which I obtained by observing a graph. Would love assistance!
0
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1answer
48 views

Why does adding the reciprocal of an equation to twice of itself give extra solutions?

Why does adding the reciprocal of an equation to twice of itself give extra solutions even though the reciprocal transformation is a one-to-one function? Is it because 2x + 1/x is not a one-to-one ...
2
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1answer
39 views

Resolving a rational system of equations with too many unkowns

A little bit of context. While working a larger proof (the proof is quite related to this question I asked), I stumbled upon the following problem. The question. Can we find $x_1,\ldots,x_{24}\in\...
3
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4answers
55 views

What is the number of elements in the solution set of $(x^2-4)^2\cdot(x^2-6x-7)=0$?

$(x^2-4)^2\cdot(x^2-6x-7)=0$ $S.S.=\{x_1,x_2,...,x_n\}$ $\Rightarrow n=?$ Answer is given as $4$. I think it should be $6$ because of multiplicity of the roots. I debated this problem with ...
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1answer
33 views

Inequality with square roots where solution found with discriminant is not valid

I have : $3 + \sqrt{x-1} > \sqrt{2x}$ when doing basic algebraic operations I will find that : $ 0 > x^{2} -52x + 100 $ I will use $b^2-4ac$ formula to eventually find out that there are ...
22
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3answers
2k views

Understanding some proofs-without-words for sums of consecutive numbers, consecutive squares, consecutive odd numbers, and consecutive cubes

I understand how to derive the formulas for sum of squares, consecutive squares, consecutive cubes, and sum of consecutive odd numbers but I don't understand the visual proofs for them. For the ...
0
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1answer
51 views

Inequality similar to Minkowski

Prove that $(|x_1-z_1|^p+|x_2-z_2|^p)^{\frac{1}{p}} \le (|x_1-y_1|^p + |x_2-y_2|^p)^{\frac{1}{p}}+(|y_1-z_1|^p + |y_2-z_2|^p)^{\frac{1}{p}}$ The form is very close to Minkowski's inequality,but I can'...
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votes
2answers
28 views

Which positive real number x has the property that x, floor of x and x - floor of x form a geometric progression

Which positive real number $x$ has the property that $x$, $\lfloor x \rfloor$, and $x - \lfloor x\rfloor$ form a geometric progression (in that order)? (Recall that $\lfloor x\rfloor$ means the ...
0
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3answers
53 views

The difference between two perfect squares Help!!! [on hold]

The difference between two perfect squares is $\ 133\ $. What is the smallest possible sum of the two perfect squares? I would like a step by step explanation. Thank you.
1
vote
1answer
39 views

Subtracting exponents properties?

Background : I was reviewing some practice problems for a small local math competition, and I don't understand how the give solution works. I don't know how this, $$9^{x+2} - 9^x=240$$ is the same ...
1
vote
5answers
74 views

Is it true that $\forall n \in \Bbb{N} : (\sum_{i=1}^{n} a_{i} ) (\sum_{i=1}^{n} \frac{1}{a_{i}} ) \ge n^2$ , if all $a_{i}$ are positive? [duplicate]

If $\forall i \in \Bbb{N}: a_{i} \in \Bbb{R}^+$ , is it true that $\forall n \in \Bbb{N} : \big(\sum_{i=1}^{n}a_{i}\big) \big(\sum_{i=1}^{n} \frac{1}{a_{i}}\big) \ge n^2$ ? I have been able to ...
0
votes
1answer
40 views

Parametric equations intuition

If I paramaterise a function like x= f(t) and y=g(t) like x=t and y=t^2 why why does eliminating the parameter give you the function that the parametric equations? It may be clear for y=x^2 but i'm ...
-1
votes
2answers
21 views