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
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2answers
130 views

How to simplify the formula for $n$th Fibonacci number when $n=2$?

When n is equal to 2 how do I simplify when the $n=2$ is put into the equation below (by the way I have to prove this formula by induction that when n= any number it will equal that number) ...
2
votes
1answer
56 views

Which of the following relations are functions of q?

Firstly, what is a function of q? Am I correct to assume it means $f(q)$? $w=q+1$ For this one, it is a linear function, so it has to be a function of q. But I'm not sure how to express it? ...
2
votes
1answer
92 views

Simplifying a square root of a square

Simplify: $$(x^2+6x+9)^{-\frac{1}{2}} \cdot (x+3)^2$$ The answer is $x+3$, but I don't understand how? There is no restriction, should it not be as follows? $$\frac{1}{\sqrt{(x+3)^2}} \cdot ...
0
votes
0answers
53 views

Proof that the $sqrt[k]{z}\, z \in \mathbb N$ counts the amount of numbers less than or equal to z with a $k-$exact power

Empirically, it can shown that $$\mathrm{Floor}[\sqrt[k]{z} ] \,, z \wedge k\in \mathbb N $$ is equal to the amount of numbers which have a $k-$exact root. For example, $\sqrt 36 = 6$ means that there ...
0
votes
3answers
79 views

Solving equations having both log and exponential forms

How can one Solve equations having both log and exponential forms: For eg... $e^x$ $=$ $\log_{0.001}(x)$ gives $x=0.000993$ (according to wolfram-alpha ...
2
votes
3answers
82 views

Explaining why $\sqrt {x^2+a} = x\sqrt{1+ \frac{a}{x^2}}$ For $x>0$.

I understand the technical operation of extracting $x^2$ out of the root, but is there a way proving it? $$\sqrt {x^2+a} = x\sqrt{1+ \frac{a}{x^2}}$$
0
votes
2answers
82 views

How to show that a given line has a certain equation?

Say line $A(3,0)$ and $B(0,2)$ How do I 'show' that they have equation $2x + 3y - 6 = 0$?
0
votes
2answers
69 views

$2x^{1/3} = 3y^{1/5}$, What is largest possible integer of $xy$? [closed]

$$2x^{1/3} = 3y^{1/5}$$ What is largest possible integer of $xy$? I need to determine the largest possible value of $xy.$ I couldn't solve this cause there is only one equation.
10
votes
14answers
2k views

Irrational numbers in reality

I have a square stone slab 1 metre by 1 metre, by the Pythagorean identity the diagonal from one corner to another is given by $\sqrt 2$. However $\sqrt 2$ is an irrational number, could someone ...
0
votes
2answers
53 views

How do I isolate $y$?

In the equation $kx - 3y = 10$, how do I isolate the $y$ component? I forgot how to do basic algebra and I am taking a calculus course, lol. I'm not sure if you multiply or subtract, I think you look ...
11
votes
3answers
685 views

How do I solve $\vert x\vert^{x^2-2x} = 1$?

I have the exponential equation $\vert x\vert^{x^2-2x} = 1$, but how do I solve it?
0
votes
1answer
21 views

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$, where $aj+b$ is a complex number, and $|f(x)|$ is the modulus function. In the past I've been calculating $|(aj+b)^{-1}|$ by multiplying the numerator and ...
1
vote
2answers
59 views

Given $(x+3)(y−4)=0$, what is the relationship between $xy$ and $-12$?

Given $(x+3)(y−4)=0 $ Quantity $A = xy $ Quantity $B = -12 $ A Quantity $A$ is greater. B Quantity $B$ is greater. C The two quantities are equal. D The relationship cannot be determined from ...
0
votes
4answers
97 views

How to solve these simultaneous equations using any better way?

This problem is easy, but I am just curious whether there is any better and more elegant method of solving. Solve the simultaneous equations: $$2x^2+5xy+2y^2=0,$$ $$x^2-y^2=1.$$ The way I solve it ...
-1
votes
2answers
75 views

Is $f(x) = 2 + \ln x$ another way to write $f(x) =\log_e x +2$?

I just want to make sure I am correctly understaning this concept. $f(x) = 2 + \ln x$ is the same as $f(x) =\log_e x +2$ Thus my T graph would look like so: e^y|x+2 -3|2.049 -2|2.135 ...
2
votes
2answers
81 views

Does loga/logb = log(a^(1/logb))?

I know $\log(a^b)=b\log(a)$. However, Wolfram Alpha tells me that $\frac{\log(a)}{\log(b)}$ does not equal $\log(a^\frac{1}{\log(b)})$. Is Wolfram Alpha correct? If it is, why is it correct? I'm ...
0
votes
1answer
73 views

How to solve a nonlinear system of three equations involving rational functions?

How do I get $a$, $b$, and $c$? Given $$X=\frac{a+\frac{1}2b}{a+b+c}$$ $$Y=\frac{b(\frac{\sqrt3}{2})}{a+b+c}$$ $$Z=\frac{76a+150b+29c}{255}$$ in other words How do i get $a$, $b$, and $c$ on the ...
2
votes
2answers
107 views

Simplify $(7-2i)(7+2i)$. Found difference between mine and solution guide's and didn't know why.

I looked up the solution guide and found out: $(7-2i)(7+2i)$ $=49-(2i)^2$ $=49+4$ $=53$ Why the unknown "$i$" just disappeared$?$ I supposed it might be: $(7-2i)(7+2i)$ $=49-(2i)^2$ $=49-4i$ does it? ...
3
votes
1answer
253 views

Solve for $f(x)$ if $f(f(x))=6x-f(x)$

If $f: [0,\infty) \rightarrow [0,\infty)$ and $f(f(x))=6x-f(x)$ $f(x)>0$ $ \forall x \in (0,\infty) $ Find f(x)
0
votes
1answer
17 views

What is the logaritmic form of $v=Ae^{Bi}$

I am reading a scientific paper, which uses a model of the form $v=Ae^{Bi}$ and then it says that this model has the following logarithmic form $\ln (v) = Bi + ln(A)$ where A is a constant. But the ...
1
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1answer
41 views

Show that $\sum _{k=1 } ^n \binom {n-1 } {k-1} x^{k-1 } (1-x)^{n-k }=1$

Show that $\sum _{k=1 } ^n \binom {n-1 } {k-1} x^{k-1 } (1-x)^{n-k }=1$ First $\sum _{k=0 } ^n \binom {n } {k } x^k (1-x)^{n-k } =1$, this follows from the binomial theorem. So this shouldn't be to ...
0
votes
1answer
42 views

Comparing Fractional Numbers

Does a formula exist for comparing two fractional numbers, without resolving to using anything other than integers and fractions? (Thus not real numbers). In other words: given $\dfrac{a}{b}$ and ...
3
votes
2answers
162 views

Simple subtraction that I can't figure out. [duplicate]

A bat and a ball cost £1.10 in total. The bat costs £1 more than the ball. How much does the ball cost? The answer to this question is somehow 5p. How?!! Should it not be 10p?
0
votes
1answer
35 views

Optics surface equation to quadric form

This should be straightforward, but honestly I forgot even the names Google for... I've got a surface description in this form (what is it called?): $$z=\dfrac{cr^2}{1+\sqrt{1-(1+k)c^2r^2}},$$ ...
4
votes
2answers
90 views

If $p(x)$ is a polynomial with integer coefficients and $p(100)=100$ what is the maximum number of integer solutions to the equation $p(k)=k^3$

If $p(x)$ has integer coefficients and $p(100)$ equals $100$ what is the maximum number of integer solutions $k$ to the equation $p(k)=k^3$. I have tried hard to solve this problem but I could not ...
2
votes
1answer
76 views

Rearranging equation with algebra

I'm having a difficult time showing that the two are equivalent: $2(x_1-\theta)(1+(x_2-\theta)^2)+2(x_2-\theta)(1+(x_1-\theta)^2) = 2(\bar{x}-\theta)(1+(x_1-\theta)(x_2-\theta))$ I have multiplied ...
0
votes
1answer
77 views

Find distance, given angles of elevation

Write an equation giving the distance d between the plane and observation post in terms of $\theta$ and $\phi$. Is this correct? when using the Law of Sines answer: $a/\sin\theta = c/\sin C$ ...
3
votes
2answers
81 views

Elementary algebra problem

Consider the following problem (drawn from Stanford Math Competition 2014): "Find the minimum value of $\frac{1}{x-y}+\frac{1}{y-z}+ \frac{1}{x-z}$ for for reals $x > y > z$ given $(x − y)(y − ...
3
votes
5answers
324 views

Calculating $\ln(1+\sqrt3)$

I distributed the natural logarithm and got $(0 + 0.549)$ [placing the values in a calculator]. However, the answer key states that the answer is $1.0051$. Where did I go wrong?
0
votes
1answer
38 views

Trouble with integration using the definition of integral

I'm playing with integration for the first time and I can understand now why everyone tells me calculus II is the hardest calculus. I'm trying to solve this problem but I think I have the wrong ...
0
votes
3answers
107 views

How do I solve the trigonometric equation $\sec^3x - 2 \tan^2 x = 2$? [closed]

A friend asked to me how could she resolve this equation, but I don't know how to resolve it?? Could you help me?. The equation is : $\sec^3x - 2 \tan^2 x = 2$ Note: She told me that I can use ...
2
votes
3answers
32 views

Feedback on solution to $30/(4x+5)=95$

The problem is $$\frac{30}{4x+5} = 95$$ I multiplied $95$ by $(4x+5)$ and then eventually got: $$-445 = 380x$$ and dividing by $380$ gives $x = -1.17$. Is this correct?
1
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1answer
64 views

Sound and decibels at distance

If I have an object that is 53 decibels at x distance, how many decibels would y objects be at the same distance x, assuming they all created 53 decibels.
1
vote
2answers
1k views

The sum of 2 consecutive numbers is 53. [closed]

The sum of 2 consecutive numbers is 53. I need to find those numbers, but I'm not even sure how to set up the problem. Any suggestions?
2
votes
1answer
52 views

Solution of equation $\frac{x\cdot 2014^{\frac{1}{x}}+\frac{1}{x}\cdot 2014^x}{2} = 2014$

Solution of equation $\displaystyle \frac{x\cdot 2014^{\frac{1}{x}}+\frac{1}{x}\cdot 2014^x}{2} = 2014$ $\bf{My\; Try::}$ Clearly Here $x>0$, Now Using $\bf{A.M\geq G.M}$ So Here ...
1
vote
3answers
76 views

Solve the equation $(x^2-9)+\sqrt{2-x}=0$

Solve the equation $(x^2-9)+\sqrt{2-x}=0$ $(x+3)(x-3)+\sqrt{2-x}=0$ Conditions: $x\neq\pm3 \wedge x\leq2$ $(x+3)(x-3) = -\sqrt{2-x}$ $(x+3)^2(x-3)^2 = 2-x$ $x^4-18x^2+81 = 2-x$ ...
0
votes
1answer
52 views

Trivial solution

What condition(s) should be met so that the equation $d^T A d=0$, where $A$ is $nxn$ symmetric matrix, has only the trivial solution ($d_1=d_2=\ldots=d_n=0$)?
1
vote
2answers
76 views

Prove equation less than 1

How do you show that $$ 2\left|\cfrac{\alpha_1-\alpha_2}{(\alpha_1-2)(\alpha_2-2)}\right|<1\qquad\text{for}\qquad0<\alpha_1,\alpha_2<1 $$ Thank you for your help and kindness.
2
votes
2answers
230 views

Rates Question (Speed/Distance/Time)

This is straight out of a maths competition. One of the few questions I can't manage to get a grip on. Luisa cycled to and from a destination using the same route. The route included some flat roads ...
0
votes
1answer
55 views

$\left | -(x+2)^2+6(x+2) \right |>13$

I did $-(x+2)^2+6(x+2)>13$ and $-(x+2)^2+6(x+2)< -13$. The first inequality had complex solutions and therefore can be disregarded but the second one has two real solutions, $x \approx -3.7$ and ...
2
votes
4answers
84 views

Is this function decreased with $x$?

Given three positive integers $a,b,c\ge 1$, I am wondering if the following $f(x)$ is decreased with $x$ ? $$f(x)=\frac{c+2x}{(a+x)(b+x)}, \quad x \in Z^+ \cup \{0\}$$ where $1\le c \le ab$.
3
votes
1answer
957 views

Factoring cubic polynomials with missing terms.

I am working on a linear algebra problem where I have to diagonalize a matrix. The characteristic equation is $-\lambda ^3 - 3 \lambda^2 + 4$. I need to factor this in order to solve part of the ...
3
votes
1answer
90 views

Solving exponential equation $e^{x^2+4x-7}(6x^2+12x+3)=0$

How would you find $x$ in: $e^{x^2+4x-7}(6x^2+12x+3)=0$ I don't know where to begin. Can you do the following? $e^{x^2+4x-7}=1/(6x^2+12x+3)$ and then find $ln$ for both sides?
-1
votes
1answer
92 views

Determine the nature of $f(x)$

Consider a polynomial $f(x)$ with real coefficients having the property $f(g(x))=g(f(x))$ for every polynomial $g(x)$ with real coefficients. Determine and prove the nature of $f(x)$. Can someone ...
1
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3answers
79 views

Help me understand a part of this answer

Help me understand the part after the second '=', what on earth is going on in the numerator. The denominator and other parts I understand.
0
votes
1answer
91 views

How to calculate a moving percentage

I need to collect a payment using a payment provider that charges 2.9% + .30 of the final payment amount. I need to ensure that a specific number is collected after the fees are taken out. I need the ...
0
votes
2answers
296 views

Projectile motion and Quadratic equation

A particle is projected from ground level so that its height above the ground after $t$ seconds is given by $(20t-5t^2)=m$. After how many seconds is it $15m$ above the ground? I do not know how to ...
2
votes
3answers
79 views

(a+b)^1/2 another question is Square root (-4)^2=?

(a+b)^1/2 and Square root (-4)^2=? I'm new to learning algebra. I know what (a+b)^2 is. But then I thought what happens with ^1/2 or ^1/4. Can someone explain me? Also I have 2 questions in my book. ...
-1
votes
4answers
195 views

Manipulating a trigonometric equation involving $\tan^2(3\theta)$ [closed]

If $\tan^23\theta = 1$, how do I manipulate the equation so I can make $\tan\theta$ the subject? I forgot how to do these since it has been a long time. I tried searching before posting. My answer is ...
1
vote
0answers
42 views

Indices Question

For these equations $n$, $m$, $p$, and $k$ are positive integers greater than $1$. $$n^{5/3}=m^{7/2}$$ $$nm=p^k$$ What is the smallest value that $k$ can be? Answer is $31$. I got as far as ...