Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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5 views

$\log \left(a^x\right)=x \log (a)$ and $-\log _x(2)=-\frac{1}{\log _2(x)}$

Since $a^x$ has domain $(-\infty ,\infty )$, $\log _a(x)$ has range $(-\infty ,\infty )$. Since $a^x$ has range $(0 ,\infty )$. Since $a^x$ and $\log _a(x)$ are inverse functions, the following ...
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1answer
24 views

Customers decrease with price increase: find the maximum price before no customers can afford candies.

If the price of the candies is $\$10$ per box, there will be $100$ boxes sold. For each $\$1$ increase in price, $5\%$ of the customers no longer can afford candies. What is the maximum price that can ...
0
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2answers
38 views

Is there a nice way to simplify this expression?

Is there a way to get a clean expression for "y" out of this? E.g. "y = ..." $y^q=a y^x+b y^z$ It seems like the most obvious thing would be to take logs, but I was wondering if there are any ...
2
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4answers
67 views

Cannot find any excluded values in this fraction

I am learning about excluded values. I am faced with this problem: $$\frac{x}{ x^2 + 9}$$ I started by trying to solve $x$ for $0$ in the denominator: $$x^2 + 9 = 0$$ $$x^2 + 3^2 = 0$$ I then ...
2
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4answers
46 views

Show that $ax^2+2hxy+by^2$ is positive definite when $h^2<ab$

The question asks to "show that the condition for $P(x,y)=ax^2+2hxy+by^2$ ($a$,$b$ and $h$ not all zero) to be positive definite is that $h^2<ab$, and that $P(x,y)$ has the same sign as $a$." Now ...
5
votes
4answers
178 views

Solving these two equations simultaneously

I'm having a hard time to solve these two equations simultaneously. I'm arriving to a very long equation.. $$x_0^2+y_0^2=(7\sqrt{2})^2=98$$ $$\sqrt{25+(x_0+2)^2}+\sqrt{4+(y_0-5)^2}=7\sqrt{2}$$
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1answer
75 views

I can't solve this equation, is it my or book's error?

I found this in my book but whatever I do I can't solve it $$(2x + 3)^2 - (1 + 2x)(2x - 1) = x^2 - (x - 1)^2 $$ it says that the solution is $\{-1.1\}$. Am I doing something wrong or not?
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2answers
39 views

How to write these quadratic equation in general form? [on hold]

Write the quadratic equation in general form: 1. $x^2=16x$ 2. $13-3(x+7)^2=0$ 3. $x(x+2)=5x^2+1$
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4answers
27 views

Converting Numbers to Exponential Form

can someone explain to me how problems such as: are converted into exponential form? I am interested in the logic and method behind this, rather than the answer. Much thanks in advance.
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0answers
56 views

Closed formula for the numbers of the form $\sqrt{1+\sqrt{4+\sqrt{9}}}$

how can i find the formula for the nth term of this series? SQ = square root $\sqrt{1}$ = 1 $\sqrt{1 +\sqrt{4}}$= sq rt of of 3 $\sqrt{1 +\sqrt{4+\sqrt{9}}}$=1.99385061 $\sqrt{1 ...
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3answers
25 views

Need help transform one Point to another.

This is probably ultra basic stuff, but I'm very rusted and don't know how to do this. Lets say I have 5 values: upperLimit $= 600$ value $= 589$ lowerLimit$ = 500$ upperLimitY $= 250 $ ...
3
votes
3answers
28 views

Writing without negative exponent

can someone please describe to me the method behind writing numbers without negative exponents such as: Maybe just show me the logic / process? Especially for number (c) because fractions really ...
4
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6answers
432 views

The process of solving the inequality $\frac{8}{19} x\ge -1$

Why did he multiply both sides by 19/8 and not 8/19 ? Is this a rule when dealing with inequalities that to remove fractions, you have to multiply by the reciprocal ?
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0answers
13 views

Polynomial Discriminant of 4*g+h^2

Suppose we have two polynomials $g,h\in \mathbb Z[x]$ with $\deg g = 2k+1 =:n$ and $\deg h=k$. As an example, take $g=x^7+2\cdot x^6+x+2$, $h=x^3+x+7$. My question is: Why does the discriminant of ...
0
votes
2answers
33 views

What is the y-cooridinate for the point on the curve with x-cooridante 20?

What is the y-coordinate for the point on the curve with x-coordinate 20? $F. 160$ $G. 162$ $H. 164$ $J. 166$ $K. 168$ The explanation says "The correct answer is G. If the x-coordinate is 20, ...
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3answers
39 views

Algebra: manipulation of expression

How does this: $$a\sqrt{2}=c$$ become this?: $$\frac{a}{c}=\frac{a}{a\sqrt{2}}$$
3
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2answers
67 views

How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?

I am reading a chapter about mathematical proofs. As an example there is: Prove that: $$(1) \space\space\space\space\space\space\space\space\space\space\space \frac{a+b}{2} \geq \sqrt{ab}$$ for ...
1
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1answer
20 views

Order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$

There is a multiple choices which syas what is the order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$? a. 1 b. 3 c. 2 d. empty I know that by considering certain cases, for example when $x<0$ or ...
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0answers
35 views

How to measure Geometric Sequences and Sums?

I have these steps as an answer to my previous question here but I can not understand the steps how from these 1+2+4+⋯+2^(m−1)+k 2^(m−1)< k ≤ 2^m 2^(m−1)+k ...
3
votes
3answers
588 views

How do you solve for x in this equation?

I tried expanding, but I still can't get rid of the exponents to isolate x. $$\frac{(1+x)^4-1}{x}=4.374616$$ Thank you in advance for your help.
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1answer
24 views

Two teams work together and alone work problem

When two teams A and B work together, it takes 18 days to complete a job. After team A works for 3 days, and Team B works for 4 days, only 1/5 of the job is done. How long does it take for Team A to ...
3
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5answers
928 views

What's wrong with solving absolute value equations in this way?

Say I have $3x-2 = |x|$. Why can't I just do this: $3x - 2 = -x$ and $3x - 2 = x$ and then get two values for $x$: $1$ and $0.5$? I know the answer $0.5$ doesn't work if you plug this in. However, I ...
0
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2answers
30 views

How to factor the quadratic $6x^2-16x=0$? [on hold]

How to factor the quadratic $6x^2-16x=0$ ? I need help solving this. I am aware of how to solve this when it is a polynomial, but I do not know how to solve with only two terms?
0
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2answers
31 views

Expressing absolute value equations as piecewise functions

I'm not sure how to express this function in piecewise form without using absolute values: $$ f(x) = 3|x-2| - |x+1|$$ I know how to do it when there is just one absolute value, such as: $$g(x) = ...
2
votes
3answers
23 views

Simplifying nested/complex fractions with variables

I have the equation $$x = \frac{y+y}{\frac{y}{70} + \frac{y}{90}} $$ and I need to solve for x. My calculator has already shown me that it's not necessary to know y to solve this equation, but I ...
2
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4answers
49 views

Writing the complex number $z = 1 - \sin{\alpha} + i\cos{\alpha}$ in trigonometric form

Now I can't finish this problem: Express the complex number $z = 1 - \sin{\alpha} + i\cos{\alpha}$ in trigonometric form, where $0 < \alpha < \frac{\pi}{2}$. So the goal is to determine both ...
4
votes
4answers
43 views

Basic understanding of Log and $2 \log _3(x)+\log _9(x)=10$

So this is what I have done so fare $$2 \log _3(x)+\log _9(x)=10$$ I know that $$\log _9(x)=\log _3\left(\sqrt{x}\right)$$ I therefore have $$\log _3\left(x^{5/2}\right)=10$$ However here is ...
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1answer
30 views

Rearranging $sk^{0.5} - \delta k = 0$ to isolate $k$

I'm going through an examination and the answer booklet and it gives the answer simply as $$sk^{0.5} - \delta k = 0$$ rearrange to $$k = (s/\delta)^2$$ Could anyone explain how to get to this ...
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3answers
33 views

Finding the variable of a coordinate point on a circle

This might be a very simple question but I am having trouble figuring it out, so if anyone can explain: A circle is marked with three points A(-3,2),...
1
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4answers
66 views

Arithmetic Mean, Geometric Mean, Harmonic Mean and their relations

If $a$ be the arithmetic mean between $b$ and $c$, $b$ be the geometric mean between $c$ and $a$ then prove that $c$ is the harmonic mean between $a$ and $b$. I expressed $a$ as $$a=\frac{(b+c)}{2}$$ ...
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1answer
64 views

Solving for $a^2+b^2+c^2$ given $\sqrt{a^2+1}+\sqrt{b^2+4}+\sqrt{c^2+9}$ and $a+b+c$. [on hold]

I am trying to solve the following problem involving three terms: $$\sqrt{a^2+1} + \sqrt{b^2+4} + \sqrt{c^2+9} = 8$$ $$a + b + c = 6$$ $$a^2 + b^2 + c^2 =?$$
0
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1answer
25 views

Arithmetic and Geometric Mean Inequalities [on hold]

Can someone help me to understand the logic of: $$\sqrt{ab} \le \frac{a+b}{2}$$ Proof: ?
1
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2answers
22 views

Finding the points where a circle intersects an axis

A circle has the equation: x²+y²+4x-2y-11 = 0 What would be the coordinates of the points where the circle intersects with the y-axis and how would you calculate it?
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2answers
47 views

How to simplify $(x+4)^2$? [on hold]

I have been getting this problem in various situations. But I always get a different answer.
0
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3answers
55 views

How to solve this word problem on the topic of quadratic equations?

The maths teacher of Mumbai is transferred to another school. The students of Class 10 decided to buy a book for 360 rupees(currency) as a gift for her. On the farewell day, 4 students did not turn up ...
4
votes
1answer
48 views

Why is this answer wrong? (rational expressions)

Simplify the following expression: $$q=\frac { z+9 }{ 5 } +10$$ this is what I got: $$q=\frac { z+9 }{ 5 } +\frac { 50 }{ 5 } $$ $$ q=\frac { z+9+50 }{ 5 } $$ This answer is wrong: $$q=\frac { ...
3
votes
4answers
77 views

If $a,b,c$ are positive, then $(a+b+c)(1/a+1/b+1/c)\ge 9$

The question asks to prove that if "$x_1,x_2,x_3$ are positive numbers show that: $$(x_1+x_2+x_3) \left(\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3} \right)\ge 9$$ I've tried to use the fact that the ...
0
votes
0answers
34 views

Need help with excel spreadsheet! [on hold]

So I am currently doing an assignment in which I have bought a house using a home-loan. For this part of the question I need to calculate how long it will take to repay the loan. So I have constructed ...
0
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2answers
48 views

how shall i find the $n$-th term of this,

How shall I find the $n$-th term of this: $\sqrt{1+2}$ $\sqrt[3]{1+2+3}$ $\sqrt[4]{1+2+3+4}$ $\sqrt[5]{1+2+3+4+5}$ $\sqrt[6]{1+2+3+4+5+6}$ $\sqrt[7]{1+2+3+4+5+6+7}$ all the way to ...
2
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0answers
55 views

$\sum_{k=1}^n \lfloor kx \rfloor =$ ?

Let $x$ be a positive real number and $n$ a positive integer , then how may we evaluate $\sum_{k=1}^n \lfloor kx \rfloor $ ? If a closed form doesn't exist then can we at least find an asymptotic ...
1
vote
4answers
58 views

If $x^n=y^n$ and $n$ is odd then $x=y$

Here, we suppose that $x,y\in\mathbb{R}$ and that $x^n=y^n$, where $n$ is odd. I want to prove that $x=y$. Maybe we can use that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1})$ So, it ...
1
vote
1answer
25 views

How do i solve this to find PMT?

I know this may seem like a stupid question but i've been up late working on this math assignment and this question just isn't working when i transpose it. So this is the formula to find Present ...
0
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0answers
16 views

Boehm's and Jacopini's insight and Math

My precalc teacher mentioned "If, then, else" in class today. So from the little programming experience I have, I picked up that the same thing exists in CS. I was wondering what that is called in the ...
0
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4answers
24 views

Progressions and ratio problem

The ratio of arithmetic mean and geometric mean of two numbers is 5:4. If the difference between their geometric mean and harmonic mean is (-0.8) find the numbers. I tried using ratio property but I ...
0
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4answers
36 views

Time it will take to complete the project if two people work together (word problem)

If Iris spends 3 days and Olivia spends 5 days on a project, 1/2 of the work can be done. If instead Iris spends 5 days and Olivia spend 3 days, then 1/3 of the work is done. How long does it take to ...
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votes
1answer
34 views

If a and b are the two solutions to x^2 - x - 2=0, then a+b=? [on hold]

If a and b are the two solutions to x^2 - x - 2=0, then a+b= ? A. -1 B. 0 C. 1 D. 3 E. 5
2
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0answers
30 views

What is the (currently) optimal root finding algorithm for multivariate functions? [on hold]

Let's say we wish to find the roots of the function: $f(x,y,\cdots) = 0 \;,$ so, for a minimal example: $xy - 1 = 0 \; .$ I know there are different methods to solve this problem for the ...
0
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1answer
16 views

$P$ is a point on a hyperbola whose focal points are $F_1$ and $F_2$. $Q$ on the line that bisects $\angle F_1PF_2$. Prove $|PF_1-PF_2|>|QF_1-QF_2|$.

$\require{cancel}$ Sorry for the grammatical mistake in the title; it was needed to keep the title under 150 characters. $P$ is a point on a hyperbola whose focal points are $F_1$ and $F_2$. $Q$ is ...
1
vote
1answer
24 views

Quotient set cardinal in $\mathbb{Z}_{12}$

In $\mathbb{Z}_{12}$ define the equivalence relation xRy if $x^2 = y^2$ Then what is the cardinal of the quotient set?
0
votes
2answers
34 views

Largest number of pairs that can be added while keeping the population at least 60% male

I'm doing problems from the AoPS Algebra Beginner's book. There's this problem that states the following, At her ranch, Georgia starts an animal shelter to save dogs. After the first three days, she ...