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
votes
1answer
60 views

If $f(x) = y$ holds, does $f(x) = x$ also hold?

What is the difference between functions: $\ y = |x| $ and $\ x = |x|$ and how these functions are represented in Cartesian Cordinate System? I think $\ y = |x| $ as capital letter V and $\ x = |x|...
3
votes
0answers
65 views

What is $\textit{the}$ discriminant of a degree $n$ polynomial?

In my high school algebra class the teacher (who is me) says that the discriminant of a quadratic polynomial $ax^2 + bx + c$ is $b^2 - 4ac$. I have read in the Wikipedia article that the discriminant ...
4
votes
1answer
127 views

Inverse of $f(x)=3^x+2^x$

I'm tring to find inverse of $f(x)=3^x+2^x$ but I don't have any clue. I tried to $$y=2^x((3/2)^x+1)$$ $$\ln y=\ln2^x+\ln((3/2)^x+1)$$ $$\ln y= x \ln2+\ln((3/2)^x+1)$$ but I can't continue
-2
votes
2answers
30 views

Find the number of roots of following equation [closed]

If $a$ is a real number,the number of roots of $\cot x- \tan x = a$ in the first quadrants.
4
votes
3answers
119 views

the derivative of $ {1\over x} + {1\over y} = 1$

I am finding the derivative of this equation, using the implicit differentiation in term of x. $$ {1\over x} + {1\over y} = 1$$ Here is what I did. $$ {1\over x} + {1\over y} = 1$$ $$ x^{-1} + y^{-1}...
1
vote
1answer
29 views

Linear Independence and Linear Dependence

I have a question that asks if 3 $3\times 1$ vectors are linearly indepedent or linearly dependent. The answer to the question finds 3 scalars, multiplies them to each vector and sets it equal to $0$. ...
2
votes
2answers
31 views

Multiplying variables with different bases and different exponents

I'm stuck. Helping kid with alg II and the instructions say to simplify the expression into one radical. $$\sqrt{10} \cdot \sqrt[4]{3}$$ I know how to do it with same base, or same exponent, but ten ...
1
vote
0answers
68 views

Finding all functions: $f(x(2y+1))=f(x(y+1))+f(x)f(y)$

Need to find all functions from integers to complex that satisfy for all $x,y \in \mathbb{Z}$: $f(x(2y+1))=f(x(y+1))+f(x)f(y)$ Any help would be great.
0
votes
1answer
51 views

Find the range of an expression

Given $$E=\frac{(1+x)^8+16x^4}{(1+x^2)^4},$$ Find the minimum and maximum possible value of $E$. I am trying AM-GM, but I am unable to break it in that format. Also, how do I calculate the lower ...
7
votes
2answers
102 views

Prove that $\frac{1}{a(a-b)(a-c)} +\frac{1}{b(b-c)(b-a)} +\frac{1}{c(c-a)(c-b)} =\frac{1}{abc}$ for all sets of distinct nonzero numbers $a,b,c$.

Prove that $$\cfrac{1}{a(a-b)(a-c)} +\cfrac{1}{b(b-c)(b-a)} +\cfrac{1}{c(c-a)(c-b)} =\cfrac{1}{abc}$$ for all sets of distinct nonzero numbers $a,b,c $. Now my question is not about how to solve ...
1
vote
1answer
77 views

If $\alpha$ , $\beta$ are roots of the quadratic equation $x^2 -2p(x-4) -15 = 0$ , then answer the following .

What is the set of of values of $p$ for which one root is less than $1$ and the other is greater than $2$ ? A) $ (7/3,\infty) $ B) $ (-\infty,7/3) $ C) $ x \in R $ D) $ None $ Please tell me ...
3
votes
3answers
106 views

If $x+y+z=6$ and $xyz=2$, then find $\cfrac{1}{xy} +\cfrac{1}{yz}+\cfrac{1}{zx}$

If $x+y+z=6$ and $xyz=2$, then find the value of $$\cfrac{1}{xy} +\cfrac{1}{yz}+\cfrac{1}{zx}$$ I've started by simply looking for a form which involves the given known quantities ,so: $$\cfrac{1}{...
2
votes
3answers
68 views

How do you evaluate this sum of multiplied binomial coefficients: $\sum_{r=2}^9 \binom{r}{2} \binom{12-r}{3} $?

We have to find the value of x+y in: $$\sum_{r=2}^9 \binom{r}{2} \binom{12-r}{3} = \binom{x}{y} $$ My approach: I figured that the required summation is nothing but the coefficient of $x^3$ is the ...
3
votes
2answers
37 views

If $ ax^2 + 2bx + c = 0 $ and $ a_1x^2 + 2b_1x + c_1 = 0 $ have a common root , then prove the following. [closed]

If $a/a_1 , b/b_1 , c/c_1 $ are in A.P. then $ a_1 , b_1 , c_1 $ are in G.P. I have no idea , how to approach this . What I have thought : For the AP series $ a/a_1 = k - d $ the rest be k &...
4
votes
1answer
87 views

Find the maximum value a sum of $6$ terms

Given $6$ non-negative reals $x_1,x_2,x_3,x_4,x_5,x_6$ such that $x_1+x_2+x_3+x_4+x_5+x_6=1$ and $x_1x_3x_5+x_2x_4x_6 \geq \frac{1}{540}$. Find the maximum value of $x_1x_2x_3+x_2x_3x_4+...
1
vote
2answers
44 views

If $ 3x^2 + 2\alpha xy + 2y^2 + 2ax - 4y + 1 $ can be resolved into two linear factors, then prove the following.

Prove that : $ \alpha $ is a root of the equation $ x^2 + 4ax + 2a^2 + 6 = 0 $. What does it mean by "can be resolved into two linear factors"? If it means $( ax + b ) ( cx + d )$ , is it necessary ...
6
votes
1answer
71 views

Find the value of $\sin(\frac{1}{4}\arcsin\frac{\sqrt{63}}{8})$

Find the value of $\sin(\frac{1}{4}\arcsin\frac{\sqrt{63}}{8})$ Let $\sin(\frac{1}{4}\arcsin\frac{\sqrt{63}}{8})=x$ $\arcsin\frac{\sqrt{63}}{8}=4\arcsin x$ $\arcsin\frac{\sqrt{63}}{8}=\arcsin(4x\...
0
votes
2answers
95 views

Exact value $\sin(160^\circ)\sin(140^\circ)\sin(110^\circ)$

I am looking for the exact value of $a = \sin(160^\circ)\sin(140^\circ)\sin(110^\circ)$. The hard way would be to compute each factor, wich is doable, and basically amounts to compute trigonometric ...
0
votes
3answers
24 views

Let $z_1$ and $z_2$ are two complex numbers such that $(1-i)z_1 = z_2$ and $ \arg(z_1z_2) = \pi/2 $ , then find $\arg(z_2)$ .

My solving so far : $\arg(z_1z_2) = \pi/2 $ $\implies \arg(z_1) = \pi/2 - \arg(z_2) $ Let $z_2 = \theta ; |z_1| = r , |z_2| = r' $ $\implies z_2 = r'\{\cos\theta + i\sin\theta\}$ & $z_1 = r\{...
-1
votes
2answers
39 views

Simplifying a log function

How come: y=c1*e^(c2*t) is simplified to: ln(y)=ln(c1)+c2*t ? What I got is: ...
0
votes
0answers
50 views

closed form solution of the following iterative equation?

is it possible to obtain a closed-form solution w.r.t. ${P_j:\forall j}$ (or in terms of special functions) for the following equations: $\frac{\lambda}{\mu}P_0=P_1$ $\frac{\lambda}{\mu}P_j=P_{j+1}+...
9
votes
2answers
133 views

$f(g(h(x)))=0$ has $8$ real roots

Find all quadratic polynomials $f(x),g(x)$ and $h(x)$ such that the polynomial $f(g(h(x)))=0$ has roots $1,2,3,4,5,6,7$ and $8$. I don't know what to do. Making a $8$ degree equation is quite tedious....
0
votes
2answers
47 views

Determine the unknown angle

For this question I'm a bit confused and don't know where to start by solving it, any guidance is appreciated.
3
votes
5answers
54 views

Remainder when $-1$ is divided by $2$

The two possible ways to find the reminder, $-1 = -1 \times 2+1$ $-1 = - 0 \times 2-1$ From the above calculation, I have found different quotients: $+1$ and $-1$. If I am asked to tell that ...
0
votes
2answers
29 views

Algebra shortcut quick method

Its a revised-Gre question it is given that $$1000 = a^2 b$$ where a and b could be any number. It is possible to find different set of answers for the equation given, but how can I know the sets ...
1
vote
2answers
93 views

Is an undefined equality vacuously true?

To quote Wikipedia, [..] equation is an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. Consider a ...
1
vote
2answers
51 views

When does $nx^4+4x+3=0$ have real roots?

Find all positive integers $n$ such that the equation $nx^4+4x+3=0$ has real roots. I think the answer must also include the cases with $2$ real roots. But my main question is, how do I start? ...
2
votes
2answers
99 views

Algebra review for Spivak Calculus

I got a bit bored with High School maths so I picked up a copy of Calculus by Spivak. I am really enjoying the book and have found that the proofs and theorems aren't as hard as others have made them ...
0
votes
3answers
21 views

Show that this section of the cone is a hyperbola

Show that section of a cone, with vertex at origin and base $x=a$ & $y^2+z^2=b^2$, intersected by a plane parallel to $XY$ axes is a hyperbola.
0
votes
1answer
33 views

Do you know the formula?

I have three data: coordinates $(x,y)$ angle in degrees ($\gamma^\circ$) distance in meters (m) How should I calculate in general a new position $(a_1,b_1)$? For example, assume the following ...
2
votes
1answer
62 views

When is sum of first $n$ natural numbers a square? [duplicate]

My question is: When is $\sum_{i=1}^n i$ a square number? I know that this means i have to solve $n(n+1)/2=m^2$. I tried with modulo 2 etc. but i don't get to it. Please help.
2
votes
2answers
144 views

Can you simplify this term?

$$X=\frac{\frac{c}{r^2}+\frac{1-c}{(1+r)^{T+1}}}{\frac{c}{r}+\frac{1-c}{(1+r)^T}-1}$$
0
votes
1answer
25 views

Solving for n. Logarithm [closed]

How come: $\dfrac{3^n}{2^{2n-1}}<0.5\cdot10^{-5}$ will be equal to $n = 45$?
2
votes
3answers
62 views

Why is $x^2 - 2x > 0$ the same as $x<0\lor x>2$

Why is "$x^2 - 2x > 0$" the same as $x<0\lor x>2$ and "$x^2 - 2x < 0$" same as $0<x<2$?
2
votes
3answers
33 views

Determine the unknown side lengths

Rounded to nearest tenth solving for $y$: tan = opposite / adjacent 12tan(53°) = 15.9 cm solving for $z$: cos = adjacent / hypotenuse 12cos(53°) = 9.6 cm Am I correct? Also, the thing that ...
2
votes
2answers
64 views

Is it possible to solve this recurrence relation?

For any real $0<x\leq1$, let $E(x)=1$. For any reals $0<a_1,a_2,\ldots,a_n\leq1$ with $a_1+a_2+\cdots+a_n\leq1$, let $E(a_1,a_2,\ldots,a_n)=1+\displaystyle\sum_{k=1}^n\dfrac{a_k}{1-a_k} E(a_1,...
1
vote
3answers
55 views

Number of values in square root in different cases

I have two equations: $x = \sqrt{16}$ $x^2 = 16$ In first case I think there will be two value of $x = \pm4$. Because $(-4) \cdot (-4) =( +4) \cdot(+4) = 16$ In the second case I am confused. It ...
0
votes
1answer
66 views

Maximum Probability to hit the bear. [closed]

A bear hides itself either behind a bush $a$ with the probability $\frac{9}{25}$ or behind bush $b$ with probability $\frac{16}{25}$ . A hunter has $5$ bullets each of which can be fired either at ...
1
vote
3answers
78 views

Can't get to solve this word problem

Price of lemon juice bottle is $4$ , price of orange juice bottle is $6$. A buyer bought $20$ bottles and the total cost is $96$. How many lemon bottles and orange bottles did the buyer get? I ...
0
votes
1answer
61 views

Can't Simplify this equation for a Ellipse(Complex Numbers)

I'm asked to sketch the set $\{z \in C : |z + i| + |z + 1| = 2\}$. I've gotten to the point where I've got the modulus form of $|z + i| + |z + 1|$: $$\sqrt{x^2+(y+1)^2} + \sqrt{(x+1)^2+y^2} = 2$$ How ...
5
votes
2answers
58 views

Given a polynomial find the minimum value of the variable.

If $x^5 - x^3 + x = a. $ Then we have to find the minimum value of $x^6$ in terms of a. The answer given is $2a - 1$ if that gives any idea. I have no idea how to approach this problem. A hint would ...
0
votes
3answers
33 views

Find the set of values of $a \in R $ for which $ x^2 + i ( a - 1 )x + 5 = 0 $ will have a pair of conjugate imaginary roots .

My solving so far : Roots of the equation are : $$ { (1 - a )i \pm \sqrt { -a^2 + 2a -21 } } \over 2 $$ Now there will be conjugate roots if -a^2 + 2a -21 > 0 , but its not . Now I have two ...
4
votes
1answer
73 views

How do I correctly solve $ \sqrt { x - 6 } - \sqrt { 10 - x } \geq 1 $ ?

$ \sqrt { x - 6 } - \sqrt { 10 - x } \geq 1 $ My solving : $x$ $\in$ $ [ 6 , 10 ] $ for both the expressions under square root to be valid . Now $$ \sqrt {x-6} > \sqrt { 10 - x } $$ since ...
1
vote
3answers
56 views

Palindromic coin toss sequence

A fair coin is tossed 8 times then find the probability that resulting sequence of heads and tails looks the same when viewed from beginning or from the end? How to approach this question because ...
5
votes
4answers
74 views

Solve the equation: $2^{2x+1}=\left(\frac{1}{32}\right)^x$

Having trouble with this problem: $$2^{2x+1}=\frac{1}{32^x}$$ Do I need to set the exponents equal to each other?
1
vote
3answers
85 views

Using Demoivre's Theorem prove that $ {\cos5 \theta} = 16{\cos^5 \theta} - 20{\cos^3 \theta} + 5{\cos \theta} $ .

$ {\cos5 \theta} = 16{\cos^5 \theta} - 20{\cos^3 \theta} + 5{\cos \theta} $ . Demoivre's Theorem $$ \{\cos \theta + i \sin \theta \}^n = \cos n\theta + i\sin n\theta $$ Where n is an integer . I ...
1
vote
2answers
30 views

If $ { z_1 - 2z_2 }\over { 2 - z_1{\bar z_2} } $ is unimodulus and $z_2$ is not unimodulus then find $|z_1|$ .

$$ \left| {{z_1 - 2z_2 }\over { 2 - z_1{\bar z_2} } } \right| $$ $$ \implies | { z_1 - 2z_2 } | = | { 2 - z_1{\bar z_2} } | $$ I dont know how to proceed now .
1
vote
1answer
30 views

Rewrite the equation $(x-a)^2 + (y-b)^2 = r^2$ to make $y$ a function of $x$

I'm trying hard to figure out how $(x-a)^2 + (y-b)^2 = r^2$ can be written as $y = b + \sqrt{r^2 - (x-a)^2}$. My book says that you’ll want to have $y$ as a function of $x$.
2
votes
1answer
68 views

About equality of nested radicals.

Allow me, please, reformulate this problem. The equal numbers $$a=\sqrt{13}+\sqrt{10+2\sqrt{13}}$$ $$b=\sqrt{5+2\sqrt3}+\sqrt{18-2\sqrt3+2\sqrt{65-26\sqrt3}}$$ have the same minimal polynomial (over $...
0
votes
2answers
74 views

How much better (as a percentage) is $A$ than $B$? [closed]

This may sound stupid, but is it correct to say $100$ is a hundred percent better than $50$ where by better I mean higher or something like that. Similarly would it be correct to ask questions ...