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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-1
votes
2answers
35 views

Squared binomial paradox?

When you square this $$(5-2)^2$$ you will get 49 $$ 5^2 - 2 * 5 * (-2) + (-2)^2$$ $$25 + 20 + 4 = 49$$ but if you do it like this (5-2) * (5-2) you will get 9 $$ 5(5-2) - 2(5-2)$$ $$25-10-10+4$$ ...
-1
votes
2answers
31 views

How to solve the quadratic equation by completing the square? [on hold]

$9x^2-18x= -3$ $4x^2-4x=1$ Thank you so much!
-3
votes
2answers
48 views

How to solve the following equation by extracting square roots?

How to solve this equation by extracting square roots? $$9x^2=36$$ $$(x-12)^2=16$$ $$(x-5)^2=25$$ $$(x+2)^2=14$$ $$(4x+7)^2=44$$ $$(x+5)^2=(x+4)^2$$
2
votes
1answer
28 views

How to solve the system of equations $y= 5x^2-2x$ and $y=10x+9 $?

I am trying to solve this system of equations $y= 5x^2-2x$ and $y=10x+9$. I have worked the problem out and I am lost could someone please explain it? I have a test tomorrow over this information.
0
votes
2answers
47 views

Factoring $x^3-8$ by grouping

I'm trying to factor by grouping. It worked for me with polynomial division, but I can't get it to work by grouping. $$x^3-8$$ The answer should be $(x−2)(x^2+2x+4)$. So first, the groups are: ...
0
votes
0answers
9 views

what is the best method to solve a large set of sparse nonlinear equation

I am trying to solve a set of nonlinear equations which is big. Indeed, newton method is not recommended since the equatios are sparse and the jacobian is not easy to inverse. Any recommendation for ...
0
votes
1answer
62 views

Integrating $\int\frac{x^2+1}{(x-1)^3(x+2)}\mathrm dx$

I am struggling with the following integral: $$\int\frac{x^2+1}{(x-1)^3(x+2)}\mathrm dx$$ I guess it all comes down to a fairly simple algebraic manipulation - but I cannot see it...
0
votes
1answer
25 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 ...
0
votes
1answer
28 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
votes
2answers
44 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
votes
4answers
81 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
votes
4answers
53 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 ...
6
votes
4answers
592 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.. $$x_0^2+y_0^2=(7\sqrt{2})^2=98$$ $$\sqrt{25+(x_0+2)^2}+\sqrt{4+(y_0-5)^2}=7\sqrt{2}$$
0
votes
1answer
85 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?
0
votes
2answers
55 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. $x(x+2)=5x^2+1$
0
votes
4answers
28 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.
-1
votes
0answers
62 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 ...
1
vote
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
29 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
votes
6answers
439 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 ?
1
vote
0answers
14 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
35 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, ...
1
vote
3answers
40 views

Algebra: manipulation of expression

How does this: $$a\sqrt{2}=c$$ become this?: $$\frac{a}{c}=\frac{a}{a\sqrt{2}}$$
3
votes
2answers
73 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
vote
1answer
32 views

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

There is a multiple choices which says 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 ...
0
votes
0answers
36 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
597 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.
0
votes
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 ...
7
votes
8answers
1k 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
votes
2answers
31 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
votes
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
24 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
votes
4answers
51 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
44 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 ...
0
votes
1answer
34 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 ...
0
votes
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
vote
4answers
67 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 $$a=\frac{(b+c)}{2}$$ ...
-3
votes
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
votes
1answer
27 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
vote
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?
-2
votes
2answers
48 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
votes
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
78 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
votes
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
votes
0answers
56 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
27 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
votes
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 ...