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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0
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1answer
13 views

Simplification a trigonometric equation

$$16 \cos \frac{2 \pi}{15} \cos\frac{4 \pi}{15} \cos\frac{8 \pi}{15} \cos\frac{14 \pi}{15}$$ $$=4\times 2 \cos \frac{2 \pi}{15} \cos\frac{4 \pi}{15} \times2 \cos\frac{8 \pi}{15} \cos\frac{14 ...
1
vote
2answers
34 views

How to express the equations as the Square root Like?

$$2\sin \frac{\pi}{16}= \sqrt{2-\sqrt{2+\sqrt{2}}}$$ What law is need to be applied here? Do I have to make the $\frac{\pi}{16}$ in a form that will be give us $\sqrt{2}$ like sin 45 degree?
0
votes
1answer
22 views

Absolute value of a complex number

Is it valid :$$ abs(a+ib)/abs(c+id)=abs[(a+ib)/(c+id)]$$ ? and also $$abs(a+ib)^m*abs(c+id)^n=abs[(a+ib)^m*(c+id)^n]$$ when i stands for imaginary number. I were trying to multiply by complex ...
4
votes
2answers
62 views

Cyclic inequality for $n$ numbers

$a,b,c,x_1,x_2,x_3,...,x_n>0, a+b+c=1,\displaystyle \prod_{i=1}^n x_i=1 $ . Prove that $$(ax_1^2+bx_1+c)...(ax_n^2+bx_n+c)\geq1$$. I've tried just writing out as a product using the product sign ...
2
votes
3answers
54 views

Number of Solutions of $y^2-6y+2x^2+8x=367$? [on hold]

Find the number of solutions in integers to the equation $$y^2-6y+2x^2+8x=367$$ How should I go about solving this? Thanks!
0
votes
2answers
56 views

Do all the properties of exponents work for every real exponent? [on hold]

I am writing an essay, but i doubt at typing that the properties of exponents works for every Real, I know it works for every Rational number. $$\forall a,n,m\in\mathbb R$$ or $$\forall a\in\mathbb ...
3
votes
3answers
126 views

Reducing “Quad Roots”

Active member of Stack Overflow who is brushing up his mathematics skills after several years of inactivity. I appreciate the help in advance! My question is pretty basic. I have having a hard time ...
2
votes
2answers
45 views

Difference of the roots of quadratic formula

I have a question to solve with roots quadratic formula that is , $$a^3 + b^3 = (a+b)(a^2-ab+b^2)$$ $$(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$ but I didn't understand how the below formula is ...
-4
votes
2answers
28 views

How to calculate a performance ratio when your goal is zero?

When it comes to performance ratio the standard answer is quite simple. Let's say for example we have a car manufacturer who aims to manufacture $1000$ cars and he just made $900$ so have here ...
2
votes
1answer
28 views

Approximating a special exponential function

I am interested in the following (modified exponential) function $f$: $$f(r) = e^{-\sqrt{r^2 + a^2}}$$ where $r$ takes on real values in $[0, \infty]$ and $a$ is a small, but non-zero, real ...
2
votes
3answers
88 views

Solving an algebraic inequality

For any $a$, $b$, and $c$ prove $$3a^2+3b^2-2b+2a+1>0$$ I tried the following $$(a+1)^2+(b-1)^2+2(a^2+b^2)-1>0\\ (a+1)^2-1+(b-1)^2+2(a^2+b^2)>0\\ (a+1-1)(a+1+1)+(b-1)^2+2(a^2+b^2)>0\\ ...
0
votes
2answers
28 views

partial fraction decomposition and balancing rule

Given the following equation: 4/(x(x + 2)) The first step is simple. Just split the denominator into its linear terms and since we have linear terms in the ...
3
votes
3answers
83 views

Proof that sum of first $n$ cubes is always a perfect square [duplicate]

I know that $$1^3+2^3+3^3+\cdots+n^3=\left(\frac{n(n+1)}{2}\right)^2$$ What I would like to know is whether there is a simple proof (that obviously does not use the above info) as to why the sum of ...
1
vote
3answers
38 views

finding the minimum slope of a tangent line

Given a tangent line that touches the function $f(x)=\frac{1}{\sqrt{18-2x}}$ at $x=a$, $(0\le a\le 7)$. Need to find the value of $x=a$ when the slope of the tangent line will be minimum. I found ...
-12
votes
1answer
38 views

Trig. function question [on hold]

Given the functions $f(x) = 3^x$ and $g(x) =\log_3x$, which of the following statements is not true? The graphs of $f(x)$ and $g(x)$ are reflections in the line $y = x$. $f(x)$ and $g(x)$ are ...
2
votes
2answers
102 views

Solving the equation $\ln(x)=-x$

I tried solving this equation for a long time but did not succeed. Any help is appreciated. $$\ln x=-x$$ I am not sure the tag is correct, I am not familiar with English mathematical terms. Please ...
5
votes
3answers
46 views

Sum of $2$ equal squares also a square

Is there an integet solution to $a^2 + a^2 = b^2$? Because there's this universift that has this logo of the pytagorean theorem where the two squares are equal, but I don't think it's possible.
1
vote
3answers
77 views

The point of contact of between two circles and common tangent at this point.

A large circle and a small circle have equations $x^2+y^2+2x-4y-27=0 $ and $x^2+y^2-12x+10y+43=0$ respectively. a) Show that the two circles externally touch at a single point and find the point of ...
1
vote
0answers
26 views

3 polynomial coefficient questions [on hold]

A general polynomial equation $a_nx^n+...+a_1x+a_0=0$, with $a_i$ real, cannot be solved in terms of the $a_i$ when $n>4$. Here are 3 questions I still have about this polynomials. $(n$ need not be ...
0
votes
1answer
32 views

nth Partial Sum $S(n) = \frac{n - 1}{n + 1}$

Given: Let: $$S = \sum^{\infty}_{n=1} a(n)$$ be an infinite series such that the nth partial sum is given by: $$S(n) = \frac{(n - 1)}{(n + 1)}$$ Find $a(3)$ $a(1):$ ...
1
vote
3answers
47 views

On finding the $n$-th term of an arithmetic progression

Given the common difference $d$, and first term $a$ (say). It is very easy to find the $n$th term of an arithmetic progression. My question is if we are given two common differences say $d_1$ and ...
2
votes
3answers
98 views

A question about eliminating square roots [duplicate]

If $\sqrt{x^2} = \pm x$, then why does $\sqrt{(x+2)^2} = x+2$ and not $\pm (x+2)$? This is driving me crazy, so feel free to elucidate. Thanks! ---EDIT--- I'm not sure how the other questions' ...
-3
votes
0answers
45 views

Find the equation of the arch [on hold]

Please help with this math question... There are two different parabolic arches ,the equation of the first arch is $y=x^2+35$ with a range of $0$ to $35$. Second arch should span $3$ times the ...
3
votes
5answers
67 views

Why doesn't squaring the radicand of a square root introduce a plus-minus sign here?

The question I have concerns the following problem: $\sqrt{4x-1} = \sqrt{x+2}-3$ $(\sqrt{4x-1})^2 = (\sqrt{x+2}-3)^2$ $\sqrt{4x-1}\times\sqrt{4x-1} = (\sqrt{x+2}-3)\times(\sqrt{x+2}-3)$ ...
0
votes
0answers
22 views

How to Create the Equation to Provide Asset Values

I really should have paid more attention in high school. I am trying to figure out how to create an equation to produce values for some assets in a mobile application I am developing. I believe it ...
0
votes
2answers
54 views

Determine if $\frac{k-1}{k}+\frac{1}{k(k+1)}=\frac{k}{k+1}$ holds

How to prove if the following equality holds? $\frac{k-1}{k}+\frac{1}{k(k+1)}=\frac{k}{k+1}$ Maybe finding a common denominator would work, but I have no idea how to do it in this example.
3
votes
2answers
45 views

Find $f(x, y)$ when $f(2x + 1, 3y -1) = 4x^2 + 9y^2 + 4x - 6y + 2$

Find $f(x, y)$ when $f(2x + 1, 3y -1) = 4x^2 + 9y^2 + 4x - 6y + 2$ I don't understand, how can we pass two things to a function? Can somebody explain what is this function, please?
0
votes
2answers
38 views

How to prove the extrema of that function $y=\sqrt{ax^2 +bx}$

$$y=\sqrt{ax^2 +bx}$$ It is known that $b \ne 0, a \ne 0$ I need to show that when $a<0$ this function has extrema and when $a > 0$ the function has no extrema (The extrema is an inner ...
1
vote
2answers
42 views

Isolate x and find derivative of function

I'm trying to isolate x from this equation: $$0.6^{x+2}-x-2 = 0$$ I know I need to use logarithms somehow and then I need to find a derivative of isolated x. Tried to use wolfram suggestions with no ...
2
votes
2answers
71 views

Why does $\frac{a-x}{x-a} = -1$

Why does $$\frac{a-x}{x-a}$$ simplify to $$-1$$? I have no idea where to start because there are multiple minus signs involved.
0
votes
1answer
44 views

Relation between root of polynomial and derivative of polynomial [on hold]

Suppose $p$ is polynomial with real coefficients.then which of the following statement is necessarily true? 1.There is no root of the $p'$ between two real roots of polynomial $p$. 2.There is ...
0
votes
1answer
12 views

Equivalent forms of exponential expressions

Rewrite the expression $27^{ t }$ as $A⋅B^{ 1−3t }$ This is a problem from KhanAcademy. Steps I took: $9\cdot 3^{1-3t}=$ $9\cdot 3^{ 1}\cdot 3^{-3t } =$ $9\cdot 3^{ 1 }\cdot (\frac { 1 }{ 3^3 } ...
5
votes
2answers
132 views

Why does the square root of a square involve the plus-minus sign?

If $\sqrt{x^2}$ can be simplified as follows: $\sqrt{x^2} = (x^2)^\frac{1}{2} = x^{\frac{2}{1}\times\frac{1}{2}} =x^\frac{2}{2} = x^1 = x$ Then why would $\sqrt{x^2} = \pm x$?
7
votes
0answers
55 views

Replacing numbers by roots of quadratic

We have $10$ numbers in the interval $(0,1)$, not necessarily distinct. At any moment, we can choose two of them, $a$ and $b$. If the quadratic $x^2-ax+b$ has two (possibly identical) real roots, we ...
3
votes
2answers
44 views

How is the plus-minus sign used for solving exponential and radical equations?

I noticed that for an equation such as $(4x+1)^2 = 289$, the plus-minus sign $\pm$ is used right after eliminating the squared expression $(4x+1)^2$ via square rooting. Yet, for an equation such as ...
1
vote
2answers
55 views

How to solve for multiple unknowns using substitution?

$R_1$, $R_2$, $R_3$, $R_4$, $R_5$ and $V_6$ suppose to be 'known' values. $$\frac{V_{n_1}}{R_1} + \frac{V_{n_1}-V_{n_3}}{R_2} + i_6 = 0$$ $$ \frac{V_{n_2}-V_{n_3}}{R_4} + \frac{V_{n_2}-V_{n_4}}{R_3} ...
2
votes
1answer
34 views

Constants for the Domain and Range of a Function

We're currently learning about this in high school, and I was looking ahead in the textbook. Since we just had finals and we're on break, I wasn't able to ask anyone about this problem. It would be ...
0
votes
3answers
42 views

Pre Algebra book Recommendation

Can anyone suggest pre algebra book for beginner. Would like to see something more than the worksheets offered online. I would prefer a book which would teach strong fundamentals concepts about ...
1
vote
0answers
38 views

proof that the expression is Real for any $z$

Please help me with this problem, I'm clueless here. $\ \ \ \ (\bar{z}+1-2i)^{1985} + (\bar{z}+1+2i)^{1985}$ $\ \ \ \ $proof that the expression is Real for any $z$
0
votes
3answers
32 views

Find a point D given points A, B, C so that ABCD is a square

Find a point $D(x, y)$ such that the points $A(􀀀3, 1)$, $B(4, 0)$, $C(0,􀀀3)$ and $D$ are the corners of a square. I am aware it is really easy but still have no idea. For example, I see that ...
3
votes
3answers
44 views

Why is $ -\sum_{i \in \text I} p_i \log_2(p_i)$ maximized for all $p_i$ equal? Is it true if $|\text I | = \infty$?

Reading a text it is stated without proof that $$ -\sum_{i \in \text I} p_i \log_2(p_i)$$ where $\sum_{i \in \text I} p_i = 1$ is maximized if $p_i$ is a constant. In the case of my theorem, the ...
16
votes
3answers
229 views

How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}…}}}}=2$ [duplicate]

How can I prove $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$$ I don't know which method can be used for this?
2
votes
1answer
29 views

Multi Variable Partial Fraction Problem

The question is to develop $$(x-\alpha)(x-\beta)(x-\gamma)\over(x-a)(x-b)$$ into partial fractions. Someone challenged me to solve this question and said the answer is ...
-3
votes
0answers
24 views

Calculation of rate of interest on monthly installments. [on hold]

In a Bankers scheme I have to pay Rs 5222 per month for a period of 10 years (120 months), Banker will return Rs 10,00,000 after 10 years. what is the rate of interest.
4
votes
1answer
79 views

Analytic solutions to a simple math trick

As proven here $3816547290$ is the only positive integer in which every digit is used; each digit is used only once; the first $n$ digits are divisible by $n$, for $n=1,...,10$. ...
8
votes
2answers
95 views

Equation with absolute values and parameters

In my textbook there is a statement that for a fixed positive integer $a$, there is at most one positive integer $b$ such that: $|b-1|+|b-2|+\cdots+|b-2001|=a(a+1)$. They claim that it's obvious but ...
0
votes
1answer
30 views

In the three angles, A, B, C of a triangle, angle B exceeds twice angle A by 15 degrees. Express the measure of angle C in terms of angle A.

In the three angles, A, B, C of a triangle, angle B exceeds twice angle A by 15 degrees. Express the measure of angle C in terms of angle A. I know it looks simple, but my reasoning does not agree ...
0
votes
0answers
24 views

Explanation on irrational equations needed.

I am solving irrational equations. In my textbook it is written that the equation $\sqrt[2k]{f(x)}=g(x)$ is equivalent to the system of equations $f(x)=g^{2k}(x)$ and $g(x)\geq0$. So if I solve the ...
4
votes
2answers
81 views

No. of different real values of $x$ which satisfy $17^x+9^{x^2} = 23^x+3^{x^2}.$

Number of different real values of $x$ which satisfy $17^x+9^{x^2} = 23^x+3^{x^2}.$ $\bf{My\; Try::}$Using Hit and trial $x=0$ and $x=1$ are solution of above exponential equation. Now we will ...
2
votes
1answer
40 views

Deriving angle from sin or cos

How can I derive the value in degrees of an angle starting from either the cos or sin value? $$ \cos(t) = c_{1} \quad \text{or} \quad \sin(t) = c_{2} $$