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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4
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4answers
1k views

Simple algebra formula for which I can't find the right answer

I have the formula $y + (z + 1) = \frac{1}{2} \cdot (z + 1) \cdot (z + 2)$, and I should work to $y = \frac{1}{2}\cdot z \cdot (z + 1)$. Somebody showed me how it's done: $y + (z + 1) = \frac{1}{2} ...
1
vote
3answers
117 views

Simple equation for $x$ but getting no proof.

Show that there is at least one real value of $x$ for which $$x^{1/3} + x^{1/2} = 1$$ I did draw the graphs of $x^{1/3}$ and $1-x^{1/2}$ and showed that they met at a point, but I don't ...
3
votes
5answers
137 views

How do I prove $\frac{ \sqrt{x+h}-\sqrt{x} }{ h}=\frac{1}{\sqrt{x+h}+\sqrt{x}}$?

$$\frac{ \sqrt{x+h}-\sqrt{x} }{ h}=\frac{1}{\sqrt{x+h}+\sqrt{x}}$$ I know I just asked a question and I did figure out how that one worked but I'm not sure how I would go about this one.
0
votes
3answers
94 views

How to prove $f(\sqrt{x}) = \sqrt{x}$?

Consider the function $f$ given by $$f(y) = \frac{y+x/y}{2}.$$ How does one prove $f(\sqrt{x}) = \sqrt{x}$? If I plug in the square root of $x$, how do I get back to $\sqrt{x}$? I'm looking over ...
2
votes
1answer
49 views

Steps to calculate, $(\frac{5}{3})^{-3}$

So first I calculate, $(\frac{5}{3})^{3}$ Which gives me, $(\frac{27}{125})$ But where should I go from there?
2
votes
2answers
897 views

Write this surd in its simplest form.

Express $\dfrac{1}{2+ \sqrt3}$ in its simplest form. NB: The textbook has the answer as $2 - \sqrt3$ but I can't see how that was achieved. I tried $\dfrac{1}{2} + \dfrac{1}{\sqrt3}$ and ...
1
vote
1answer
42 views

Equation of a tangent line to a curve

Equation of the curve is $y=(x+9)/(x+5)$, we are looking for the tangent line to that curve that also goes through $O(0,0)$. Answer given is $x+25y=0$ which I found to be true for $A(-15, 3/5)$ being ...
1
vote
1answer
215 views

Is $\pi = 4$ really? [duplicate]

Can anyone explain what's wrong with this?
3
votes
2answers
76 views

If you were asked to evaluate $x^2$ for $x = -1$

Would you bracket the $x$? I ask this because $-1^2$ is equal to $-1$, but $(-1)^2$ is equal to $1$. Which is valid?
2
votes
3answers
136 views

If $f(x) = 3x-4$ (functions, highschool)

if $f(x) = 3x - 4$, find $x$ when $f(x) = 7$. I would show my working out, but I have never experienced this type of question, nor have I been taught how to do it.
5
votes
4answers
203 views

Solve $x^{3}-3x=\sqrt{x+2}$

Solve for real $x$ $$x^{3}-3x=\sqrt{x+2}$$ By inspection, $x=2$ is a root of this equation. So, I squared both sides and divided the six degree polynomial obtained by $x-2$. Then I got a ...
0
votes
2answers
48 views

If $ f(x) = \sin x+\lfloor \frac{x^2}{a}\rfloor $ be an odd function. Then the set of values of parameter $a$ is/are

Let $f: \left[-10\;,10\right]\rightarrow R\;,$ where $\displaystyle f(x) = \sin x+\lfloor \frac{x^2}{a}\rfloor $ be an odd function. Then the set of values of parameter $a$ is/are $\bf{options::}$ ...
0
votes
0answers
38 views

If $X = \{a_{1}\;,a_{2}\;,a_{3}\;,\ldots,a_{6}\}$ and $Y = \{b_{1},b_{2},b_{3}\}$. Then no. of function from $X$ to $Y$

Let $X = \{a_{1}\;,a_{2}\;,a_{3}\;,\ldots,a_{6}\}$ and $Y = \{b_{1},b_{2},b_{3}\}$. Then no. of function from $X$ to $Y$ such that it is onto and there are exactly three elements in $X$ such that ...
1
vote
1answer
62 views

Is my working correct? Exponenial decay

Is my working correct? If not, please let me know where I have gone wrong. Thank you for taking the time to check! Question: A thermometer that has been stored indoors where the temperature is 22 ...
0
votes
2answers
89 views

solve the equation for x

Hi i have a homework to solve an equation for x and i have been trying to solve this for an hour and got confused. Please help me with it and Thank you. it would be great if it is step by step. $-4 x ...
0
votes
4answers
52 views

Showing that these two lines are parallel.

$$ \dfrac{x - 1}{2} = 2 - y = 5 - z \quad \text{and} \quad \dfrac{4 - x}{4} = \dfrac{3 + y}{2} = \dfrac{5 + z}{2}. $$ I was given this math problem as homework, and I have spent the past hour ...
1
vote
2answers
65 views

Cannot solve by hand:$ x + y = 2; 4x^2 + y^2 = 5(2x - y)(xy)^{\frac12}$

Firstly, this is not my homework. I am well past high school (finished graduate school several years ago) but I am mentoring a high schooler and I want to explain how to solve this by hand using just ...
1
vote
1answer
77 views

Finding the intersection of a circle and a line

The text says: On a single set of coordinate axes, sketch the line $x+16 = 7y$ and circle $x^2+y^2-4x+2y=20$ and find their points of intersection. Hint: eliminate x algebraically and solve the ...
1
vote
5answers
78 views

Why does $\frac{(x^2 + x-6)}{x-2} = \frac{(x+3)(x-2)}{x-2}$?

I'm not the best at algebra and would be grateful if someone could explain how you can get from, $$\frac{x^2 + x-6}{x-2}$$ to, $$\frac{(x+3)(x-2)}{x-2}$$
0
votes
2answers
163 views

Finding the domain of $f(x)=\ln(3x-4).$

I am trying to find the domain of $f(x)=\ln(3x-4)$. I cannot find out how to get the domain. but I did manage to get the vertical asymptote which is $x=4/3$.
1
vote
3answers
214 views

Tangents of circles

I'm trying to solve the following problem: Find the tangent equations of $x^2 + y^2 = 1$ which pass though point $(1, 2)$. As a line which goes though the point $(1, 2)$ is in the form $y = m(x - 1) ...
12
votes
5answers
2k views

Why does two terms immediately adjacent “mean” multiply?

I am currently teaching a GED math class. While learning about the order of operations, the students asked why does a number next to a parentheses mean multiplication? I understand the rule that two ...
1
vote
1answer
62 views

A simple function equation in calculus-1 course

Here is a homework question: $f^2(\ln x)-2xf(\ln x)+x^2\ln x=0,\ f(0)=0,\ f(x)=$? I don't know how to solve it. Thanks!
1
vote
1answer
61 views

Multiplication problem

I was solving the one physical numerical during which i came through a calculation $$1255\times\left(\frac{170.474}1\right)\times\left(\frac{1000}1\right)\times\left(\frac{1}{48.26}\right)^3$$ ...
4
votes
1answer
155 views

Find a non constant function that is a quotient of two polynomial, for which: $f\left(x+\frac{1}{f(x)}\right)=f(x)f(-x)$

Before I post the problem, I want to ask what is wrong with the exactly same problem I posted three days ago, 'cause nobody seemed willing to answer it. The non constant function must satisfy the ...
4
votes
6answers
140 views

Finding the coefficient on the $x$ term of ${\prod_{n = 1}^{20}(x-n)}.$

I am trying to find the coefficient on the $x$ term of $\displaystyle{\prod_{n = 1}^{20}(x-n)}$. The issue is that the binomial theorem can't be applied since our $b$ value is changing from term to ...
1
vote
4answers
458 views

Ordered pair satisfying the equation $x^2 + 6x + y^2 = 4$

How many ordered pairs of integers $(x,y)$ satisfy the equation $x^2+6x + y^2 = 4$? I can't approach this problem. Please help me.
-1
votes
1answer
28 views

Oscillating Spring & Rates of change

How to solve? Are they asking for: instantaneous rate of change: $\frac{d}{dt}h(t)=2.5$ and solve for value of $t$ or when $\frac{d}{dt}h(t_1)$ where $t_1$ is when $h(t)=2.5$ but both methods ...
4
votes
3answers
128 views

An interesting equation involving many iterations.

Let $$f(x) = 1 -|1- 2x |.$$ Find the number of solutions of the equation $$f ( f ( f ( f ( f ( f ( f ( f ( f ( f (x))))))))))=x,$$ i.e., $f^{(10)}(x)=x$. And what about if there is an arbitrary ...
0
votes
1answer
24 views

$\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$?

I want to find out if this affermation is true: let $f\in \mathbb{Q}[x]$ such that $\gcd(f,f')=1$ Does this imply that f has not multiply irreducible factors in $\mathbb{C}[x]$? (We know that it has ...
0
votes
1answer
53 views

Was my reasoning correct for this?

So I was working on this question which states: Let $x$, $y$, and $z$ be positive real numbers which satisfy the equation: $$ xy + yz + zx = xyz(x+y+z) $$ What is the maximum value for ...
7
votes
2answers
273 views

Finding all possible values of $x^4+y^4+z^4$

Given real numbers $x,y,z$ satisfying $x+y+z=0$ and $$ \frac{x^4}{2x^2+yz}+\frac{y^4}{2y^2+zx}+\frac{z^4}{2z^2+xy}=1$$ Find all possible values of $x^4+y^4+z^4$ with proof. My attempt : Putting ...
0
votes
2answers
106 views

Intro to Discrete Math: compound interest calculation

The following is from an intro to discrete mathematics page. It's on compound interest. http://www.cs.odu.edu/~toida/nerzic/content/intro2discrete/intro2discrete.html[1] Scroll to the part with the ...
1
vote
1answer
28 views

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$. [duplicate]

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$. I tried to separate the terms first and I got $\dfrac12 (\log(1+\log x) - \log(1-\log x))$. The answer is $\dfrac1{x(1-\log x)^2}$.
3
votes
3answers
98 views

Values of $n$ for which $\lfloor 2 x\rfloor +\lfloor 4 x\rfloor +\lfloor 8 x\rfloor +\lfloor 20 x\rfloor =n$ has a solution

$$\lfloor 2 x\rfloor +\lfloor 4 x\rfloor +\lfloor 8 x\rfloor +\lfloor 20 x\rfloor =n$$ How would you find the values of $n$ for which the equation has a solution under the condition that $n \leq ...
0
votes
2answers
78 views

Trigonometric identity problem

$$\frac{\sin(2x)}{2 \sin (x)}-\frac{\cos(2x)}{\cos(x)+\sin(x)}=\sin(x).$$ I got it on a test and want an answer. I always hit a dead end with the identities I learned.
8
votes
3answers
544 views

Simplifying $\sqrt[4]{161-72 \sqrt{5}}$

$$\sqrt[4]{161-72 \sqrt{5}}$$ I tried to solve this as follows: the resultant will be in the form of $a+b\sqrt{5}$ since 5 is a prime and has no other factors other than 1 and itself. Taking this ...
2
votes
3answers
51 views

Is the equation before applying additive and multiplicative properties the same equation as the equation after application?

When solving equations, one can use the additive and multiplicative properties of equality to transform a true equation, but what is the relationship between the original equation and the new ...
4
votes
3answers
54 views

Solving system $\{x+xy+y=223,~x^2 y+x y^2=5460\}$

$$x+xy+y=223$$ $$x^2 y+x y^2=5460$$ I need to find the integer solutions to this equation. However, from the looks of it a simple substitution and solve will be difficult, so it seems that clever ...
1
vote
6answers
145 views

Is there a proof that $ x^2 > (x-y)(x+y)$?

I noticed that when I did the math, $5^2 > 4 \cdot 6$, and $10^2$ is greater than $9 \cdot 11$, etc. I looked for a proof of this and couldn't find one. Assuming $x$ and $y$ are real numbers and $y ...
1
vote
3answers
56 views

How to prove that $ \sum_{n=0}^\infty \frac{1}{(2n+1)^2} + \sum_{k=1}^\infty \frac{1}{(2k)^2}=\frac{4}{3} \sum_{n=0}^\infty \frac{1}{(2n+1)^2}$

How to prove $$ \sum_{n=0}^\infty \frac{1}{(2n+1)^2} + \sum_{k=1}^\infty \frac{1}{(2k)^2}=\frac{4}{3} \sum_{n=0}^\infty \frac{1}{(2n+1)^2}$$
1
vote
1answer
44 views

Quadratic inequalities involving two solutions

I have a quadratic expression, which I have factored to correctly be: $(x-9)(x-2) > 0$ However, I don't know how to determine the two values of X after this, the correct answers are x < 2, x > ...
0
votes
5answers
94 views

How do you factor a quadratic expression, without using the formula?

I am asked to factor $2x^2 -3x+1=0 $ using factorization, but I run into fractions, and it becomes very messy and complicated to deal with, especially since specifically asked not to use the formula. ...
0
votes
3answers
195 views

“Write an equation of line J, that passes through P, and is parallel to given line L”

I apologise if this is too simple a level for this board. The question is: "Given the point $P(x_1,y_1)$ and the line $l$ with the equation $ax + by = c$: Write an equation of the line $j$ that ...
0
votes
3answers
127 views

Question about solving systems of equations (Highschool level)

If I am asked to solve a systems of equation, how would I know which method (substitution, or elimination) to use? What set of conditions should I be looking for, or is it that either method should in ...
0
votes
2answers
55 views

$x^x-x+5=\frac{29}{4}$

A friend of mine is claiming to have a closed form solution to $x^x-x+5=\frac{29}{4}$, plotting it into wolfram alpha gives an approximation, and the equation doesn't seem very easy to solve. Can any ...
2
votes
3answers
105 views

Simple question (hopefully) on unitary method

In India we have an exam called NEST. I gave it today, and this was a question I encountered: Lactobacillus sp. and Streptococcus sp. are two bacterial species responsible for curdling milk. One ...
-1
votes
4answers
85 views

Why do you add +- to only one side when you remove square root from both sides?

As the title says, why when you take a square root of both sides of the equation do you add $\pm$ only to the side which is a number, as opposed to an unknown? For example: $$x^2 = 9 \implies x = ...
1
vote
1answer
113 views

Extraneous solutions.

I just learned of extraneous solutions on the internet and thought, "could you both lose and gain solutions in one equation?" I think that, yes, you should be able to do that. However I haven't been ...
0
votes
1answer
53 views

Question about solving systems of equations

Is their a universal method to solve systems of equation, eg do methods such as 'elimination' work for ALL types of simultaneous equations (I am specifically referring to 2 and 3 equation simultaneous ...