# Tagged Questions

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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### How is the discriminant is able to find coefficients in a quadratic equation?

I know how to solve for $k$ in $kx^2-30x+25=0$ using $b^2-4ac$, but i want to know how the discriminant does this. How are we able to just plug the coefficients into the discriminant and get the ...
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### Solving for Variable in Rational Expressions

Hi I have a really simple issue, I am doing a circuit problem and I get a final equation: $$\frac{v_1-12}{2\ k\Omega}+\frac{v_1}{k\Omega}+\frac{v_1}{r_1}=0$$ Im trying to get $v_1$ by itself, the ...
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### Prove that $\pi > 24\small{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}}$

Prove that $\pi > 24\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}$. I tried using trig but I couldn't solve it. A hint I was given is to use half angle identities. This should be easy for someone who is ...
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### Solving functional equation $f\left(\sum_{i=1}^n a_i^n\right)=\frac{1}{k} \sum_{i=1}^n f(a_i^n).$

Given natural number $n, k$ consider nondecreasing function $f:\mathbb{N}\cup {0} \to \mathbb{N}\cup {0}$ such that $$f\left(\sum_{i=1}^n a_i^n\right)=\frac{1}{k} \sum_{i=1}^n f(a_i^n),$$ for ...
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### Why isn't $f(x)=\sqrt{2-x}$ reflected across the y-axis?

If I try to graph this function, it does not appear to reflect across the y-axis when it comes time to do the reflection. Rather, it is reflected around the point where the function begins on the ...
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### Looking for resources for understanding derivation of demand from utility

I am struggling with my homework and would very much appreciate a rundown of the math or pointers to where I can find help otherwise. Quoting from the assigment: There are $n$ sectors in the ...
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### $2k^2+7k=2mk+3m+36$. Find all non-negative integer solutions.

I've tried this: $2k(k-m)+7k-3m-36=0$. And I'm stuck. How do I solve this one?
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### Trig sum: $\tan ^21^\circ+\tan ^22^\circ+…+\tan^2 89^\circ = ?$

As the title suggests, I'm trying to find the sum $$\tan^21^\circ+\tan^2 2^\circ+...+\tan^2 89^\circ$$ I'm looking for a solution that doesn't involve complex numbers, or any other advanced branch in ...
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### Exponential equations with variables on both sides

I have the following: $$8^{3x+4} = 5^{4x-2}$$ How would I solve this? I tried this: $$(3x+4)\log 8 = (4x-2)\log 5$$ but have no idea where to go from there. Thank you!
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### Trigonometric equation $\sin(60^\circ-2X)\sin(5X)=\sin(8X)\sin(3X)$

A trigonometric equation is to be solved, the solution ($X=10^\circ$) is very clear but I need a proper method $$\sin(60^\circ-2X)\sin(5X)=\sin(8X)\sin(3X).$$
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### Multiplying a factorial with non factorial

I'm trying to understand the following equation, do I multiply the 2 and the 1 to get (n+2)! ? $$(n+2)(n+1)! = (n+2)!$$
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### $x^2-xy-2x+3y=11$. Find natural solutions.

I've got this factorizing: $(x-2)(x-y)=11-y$. And I'm stuck on it. How can I solve it?
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### Adding and Subtracting numbers with exponents

Why is $2^{k+1} + 2^{k+1} = 2^{k+2}$ and not $4^{k+1}$
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### $2x^2+2xy-x+y=112$. Find natural solutions.

I've got this and I'm stuck: $(x-2)(x-y)=11-y$. Is it possible to make something out of this? Oops, I've got that on the other problem.
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### If $1(0!)+3.(1!)+7(2!)+13(3!) +21(4!) + \cdots$ n terms… [closed]

Question( from sequences) : If $1(0!)+3.(1!)+7(2!)+13(3!) +21(4!) + \cdots$ n terms = $(4000)(4000!)$ Then what is the value of n. How to proceed in this please suggest , will be of great help to ...
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### What is the sum $\sum_{r=1}^\infty \frac{r}{4r^4+1}$ equal to?

Problem : If $$T_r =\frac{r}{4r^4+1}$$ then the value of $$\sum^{\infty}_{r=1} T_r$$ is ? How to start such problem I am not getting any clue on this please suggest thanks .
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### The equation $5^x+2=17^y$ doesn't have solutions in $\mathbb{N}$

Problem: Prove that the equation $5^x+2=17^y$ doesn't have any solutions with $x,y$ in $\mathbb{N}$. I've been analyzing the remainder while dividing by $4$, but I'm getting nowhere.
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### How would I be able to tell if some vector is in the span of a set of vectors?

Given the following, how would I be able to tell if b and c are in the span of the set of vectors S? Any help is appreciated. enter link description here
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### A simple problem of the equation of a plane.

Two planes given $$x-y+z=5 , \hspace{0.5cm}x+y+z=3$$ Their intersection is a line $l$.Find the equation of a plane such that the line $l$ is perpendicular to that required plane and this plane ...
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### How to show that if average of squares equals square of average then all $X_i$ are equal?

Defining $A(X) = \sum_{i=1}^np_iX_i$, how would I show that given $A(X^2) = A(X)^2$, all $X_i$ must be equal? I tried by contrapositive - assume they are not equal and show that $A(X^2) \ne A(X)^2$. ...
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### Show that $\gcd(a,b)>1$

Given are three natural numbers $a$, $b$ and $c$, for which $$\frac1a+\frac1b=\frac1c,$$ show that $\gcd(a,b)>1$. Could you someone provide a hint? I already tried algebraic manipulation, but ...
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### integer $n$ for which $n^6+3n^5-5n^4-15n^3+4n^2+12n+3$ is a perfect Square

Prove that the no integer $n\;,$ for which $n^6+3n^5-5n^4-15n^3+4n^2+12n+3$ is a perfect Square. My Try:: We can write $(n^6+3n^5-5n^4-15n^3+4n^2+12n+3) = (n^3+an^2+bn+c)^2$ Now Here we have to ...
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### How do I solve for x and y: x + 0.0467 + y = 1.000?

I am trying to find the isotopes for percent abundance question. I am looking over and answer and can't figure it out because of the math. Here it goes. 1) Set up a system of two equations in two ...
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### find minimum and justify it

After having found the derivative of which if i am not mistaken is : I need to find the minimum of the function for which if I am not mistaken I equal to 0 the first derivative is this the right ...
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### Partial fraction help

I need Help figuring out how to solve the indefinite integral of $$\int{ -5x^3-2x^2+32\over x^4-4x^3 } dx$$ using partial fractions. Please help. Thank you! I have already checked the online ...
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### Value of $x$ where the graph lies below the graph of f(X)

From this question (How can I find the values of $x$ where a function lies below or above the axis?) I learn that "lies below" means $f(x)<0$ , now my question is , how can I check values that lie ...
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### How can I find the values of $x$ where a function lies below or above the axis?

Let's imagine this problem: Find the values of $x$ where the graph of $$f(x)= \frac{3x^2}{x^2-1}$$ lies below the $x$-axis. I know how to find the intercept $(0,0)$, but I don't understand what ...
### no. of mapping from from $A\rightarrow B$ such that $f(i)<f(j)\;\forall \; i<j,$ is
If $A = \left\{1,2,3,4\right\}$ and $B = \left\{1,2,3,4,5\right\},$ Then $(a)\; ::$ Total no. of mapping from from $A\rightarrow B$ such that $f(i)<f(j)\;\forall \; i<j,$ is $(b)\;\;::$ ...
### Equation $\sqrt[n]{a_1}+\sqrt[n]{a_2}+\cdots+\sqrt[n]{a_k}=\sqrt[n]{b_1}+\sqrt[n]{b_2}+\cdots+\sqrt[n]{b_l}$
Let $k,l$ be natural numbers and $\{ a_i, b_i \}$ be real positive numbers such that $a_1\leq a_2 \ldots \leq a_k$, $b_1\leq b_2 \ldots \leq b_l$ and  \sqrt[n]{a_1}+\sqrt[n]{a_2}+\cdots+\sqrt[n]{...