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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5answers
4k views

How can I solve $\cos^2 x + \sin x +1 = 0$?

The solution set of the equation $$\cos^2 x + \sin x +1 = 0$$ is? I haven't studied trigonometry, I'm kinda lost on this issue ...
0
votes
2answers
339 views

Geometric Progression prove question

Given that a, b, c and d are in geometric progression prove that: $(b-c)^2 + (c-a)^2 + (d-b)^2 = (a-d)^2$ I've established that ar = b, br = c, cr = d where r is a common ratio. However I do not ...
0
votes
2answers
2k views

The ratio of boys to girls in a certain classroom was 2:3. if boys represented …

The ratio of boys to girls in a certain classroom was $2:3$. If boys represented five more than one third of the class, how many people were there in the class room? I do not seem to get how to solve ...
3
votes
2answers
73 views

Number of solutions of exponential equation

Can anyone tell me how to find number of solutions $(x+a)^x=b$? For example $(x+1)^x=-1$ has four complex solutions, $(x+3)^x=10$ has two solutions,one positive one negative, and $(x-4)^x=-10$ hasn't ...
4
votes
3answers
186 views

Stuck on Square Root Problem (yep, homework!)

Here's the simple question: Devon has a piece of poster board 45 cm by 20 cm. His teacher challenges him to cut the board into parts, then rearrange the parts to form a square. a) What is the side ...
1
vote
4answers
100 views

How do I factorise this polynomial

Please help factorise this: $$6x^2+x+4=0$$ In my attempts, I assumed $a=6$, $b=1$ and $c=4$. I multiplied $c$ by $a$ and attempted to get the factors that give us the sum of $b$. The only factor ...
2
votes
3answers
287 views

Proving $f(x)=(1/2)(x+(1/x))$ is not one to one

Let f : (0,∞) → R be defined by $f(x):=\frac{1}{2}(x+\frac1x))$. Prove that $f$ is not one to one (injective). I understand the usual procedure for this would be to assume that $f(x)=f(y)$ implies ...
3
votes
1answer
62 views

How do I get the equation of the line given this slope and graph?

This might be really low level for you guys but I just don't understand.! The graph is $y=1/(x-1)$ and the given slope is $-1$. I have to get the equation of the tangent line given the slope and ...
3
votes
2answers
502 views

Find the limiting value of the sequence

A sequence is given by the recurrence relation: $$u_n = 1 + {1\over u_{n-1} +1}, u_1 = 1, n{=\ge}1$$ Work out the 2nd, 3rd and 4th term of the sequence and find the limiting value of the sequence. ...
0
votes
3answers
232 views

Simplifying this algebraic expression

I'm finding algebraic holes in my knowledge. Show how you simplify $$\frac{r-r^3}{1+r}$$ into $$r(1-r)$$ and please show every step.
1
vote
3answers
498 views

What is the difference between the domain of a variable and the domain for an equation?

domain of a variable: The given set of numbers that the variable may represent. What does it mean for an equation to have a domain? Suppose the domain for an equation is {1, 2, 3, 4, ...} How is ...
3
votes
1answer
247 views

Aptitude Question

An Engine length $1000 $ m moving at $10$ m/s. A bird is flying from engine to end with $x$ kmph and coming back at $2x$. Take total time of bird traveling as $187.5$ s. Find $x$ and $2x$. My ...
1
vote
1answer
125 views

Solve the equation with logs?

I need to solve this equation $$3^{x+1} =15$$ with logarithms. If you can, could you please explain thoroughly?
1
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2answers
80 views

Basic vectors question..

If a car was traveling with velocity v, then 5 minutes later its velocity was 1.1v and then 10 minutes later its velocity was -v. What happened? Am I right in thinking that the car sped up and then ...
1
vote
1answer
344 views

Find the $n$th term of the series, where the sum of the first n terms of the series is $(4n + 5)^2$

The sum of the first $n$ terms of the series is $(4n + 5)^2$. Find the $n$th term of the series. So far I have got $n_2=88, n_3=120, n_4=152, n_5=184 ...$ How should I continue?
2
votes
2answers
329 views

Closed-form Geometric series of both increasing and decreasing variables?

This question comes from the formula $$x^n - a^n = (x-a)(x^{n-1}a^0 + x^{n-2}a^1 + .... + x^1a^{n-2} + x^0a^{n-1})$$ which can be verified by summing the second factor as a geometric series. My ...
2
votes
8answers
325 views

A function such that $f(0) = 1$ and $f(x) = x$ otherwise

Let $x$ be a real number and $f$ a function such that $f(x)=x$ if $x\not=0$ and $f(0)=1$. Does there exist a function like this, with an algebraic formula? EDIT: Thank you for your answers, I know ...
3
votes
3answers
307 views

Find time when 2 cars meet?

There is a roadway between city A and B . A car P starts at 5:00 am from A and reaches B at 10:00 am. Another car Q starts from B at 7:00 am and reaches A at 9:00 am. Find the time when car P meets ...
-3
votes
1answer
100 views

Elementary Algebra question

Ok guys so I've been out of school for eight years, never used algebra again, also I was forcibly removed from school in 9th grade. I need to ask a few questions on Elementary Algebra, what are the () ...
11
votes
1answer
331 views

Finding $x^4 + y^4 + z^4$ using geometric series

This is a problem from the 2001 Stanford Math Tournament Algebra section. $$$$Given that $$x+y+z=3$$ $$x^2 + y^2 + z^2 = 5$$$$x^3+y^3+z^3=7$$Find $x^4+y^4+z^4$. $$$$My friend claimed that he was able ...
1
vote
4answers
504 views

Limit of a Recursive Sequence

I'm having a really hard time finding the limit of a recursive sequence - $$ \begin{align*} &a(1)=2,\\ &a(2)=5,\\ &a(n+2)=\frac12 \cdot \big(a(n)+a(n+1)\big). \end{align*}$$ I proved ...
1
vote
1answer
148 views

Is an algebraic formula to test real numbers equality?

Is there a formula to test numbers equality ? Let $x$ and $y$ real numbers. If $x=y$ the formula will results $1$. Else the formula will results $0$. I'm not searching for an algorithmic solution ...
1
vote
2answers
418 views

Find the 12th term and the sum of the first 12 terms of a geometric sequence.

A geometric series has a first term $\sqrt{2}$ and a second term $\sqrt{6}$ . Find the 12th term and the sum of the first 12 terms. I can get to the answers as irrational numbers using a calculator ...
1
vote
0answers
80 views

Property of natural numbers

Can every integer be written as $$N=\prod _{ n=1 }^{ n=k }{ { (a }_{ n } } { x }_{ n }+{ b }_{ n }{ y }_{ n })$$ where 1)$(a_n,b_n )$are coprime 2)$( a_n,b_n,k)$ are any (random) integers 3)$( ...
4
votes
1answer
176 views

Find $x$ in the equation $ax^3+bx^2+cx=d$

$\begin{equation} \tag{A} ax^3+bx^2+cx=d \end{equation}$ We can define Delta for quadratic equation to check whether the equation has answer or not....for $f(x)$ which contains powers higher of $2$ ...
6
votes
2answers
341 views

Solve the equation for x, y and z: $\sqrt{x-y+z}=\sqrt x - \sqrt y + \sqrt z$

I am having some trouble with this problem, Solve for $x,y,$ and $z$. $$\sqrt{x-y+z}=\sqrt x - \sqrt y + \sqrt z$$ Here is my work so far, $$x - y +z = x+y+z-2\sqrt{xy} + 2\sqrt{xz}- ...
-1
votes
1answer
84 views

If $a,b,c \in R$ such that $c \neq0$ If $x_1$ is a root of $a^2x^2+bx+c=0, x_2$ is a root of $a^2x^2-bx-c=0 $ and $x_1 > x_2 >0$…

Problem : If $a,b,c \in R$ such that $c \neq0$ If $x_1$ is a root of $a^2x^2+bx+c=0, x_2$ is a root of $a^2x^2-bx-c=0 $ and $x_1 > x_2 >0$ then the equation $a^2x^2+2bx+2c=0$ has roots $x_3$ ...
8
votes
4answers
245 views

equilateral triangle; $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 + c^2 + d^2)^2.$

In equilateral triangle ABC of side length d, if P is an internal point with PA = a, PB = b, and PC = c, the following pleasingly symmetrical relationship holds: $3(a^4 + b^4 + c^4 + d^4) = (a^2 + b^2 ...
2
votes
2answers
202 views

Is there a name for this observation involving single-variable limit to infinity?

Obviously, $$\lim_{n\to \infty} \sqrt{n^2 +1} = \infty$$ However, the following (I am sure) is true: $$\lim_{n\to \infty} \sqrt{n^2 +1} = n$$ Is there a name for this kind of behavior in limits? ...
1
vote
1answer
127 views

Function for cosine transformation around $\pi/2$

Given the cosine of an angle $x$ relatively close to $\pi/2$, is there a function $f$ such as: $f(cos(x))=cos((x+\pi/2)/2)$ ?
1
vote
3answers
232 views

Simple integration (area under the curve) - help

I'm currently doing a simple integration question: Here is my working/solution so far: I have calculated this several times and only be seem to be getting a negative number as the final result. ...
1
vote
2answers
98 views

What is the opposite of ^?

I have been given this: $\text{XP} = (\text{level}^3)\cdot10$ So for example 270 XP = Level 3 My problem is I already have the value of XP, so how can I figure out what level would be? Thanks in ...
-1
votes
3answers
82 views

Simplify this fractional expression

Simplify this fractional expression $$\frac{4b}{3y} \frac{-4y}{12b^2} $$ This is the format in which the question is written and I do not understand what to do in order to solve for.
0
votes
1answer
31 views

Iteration problem for $a$ and $b$

I have a problem here with Iteration question: $$ f(t)=2 \sec⁡t+2t-3,\;\text{where }t=0.4\text{ and }t=0.5. $$ Now I need to show that the equation $f(t)=0$ can be rearranged to give the iterative ...
1
vote
2answers
119 views

System of quadratic equations

How would you solve the following system of equations: $$ x^2 + y = 4 \\ x + y^2 = 10 $$ Thanks very much! I tried defining y in terms of x and then inserting in to the second equation: $$ y = 4 - ...
0
votes
3answers
53 views

Find the value of $a$ if the distance between $(3,-2)$ and $(4,a)$ is $\sqrt{7}$

Find the value of $a$ if the distance between $(3,-2)$ and $(4,a)$ is $\sqrt{7}$. Do I use the distance formula with the variable or not?
2
votes
0answers
323 views

Cubic roots and Cardano formula

On solving the cubic equations, applying Cardano formula yield complex results. I wanted to evaluate the exact roots (not numerical) but I ended up with complex numbers/nested radicals. To get rid ...
4
votes
1answer
130 views

Problem 2-7 in Spivak

One is asked to show that $ \sum\limits_{i=1}^{n} k^{p}$ (typo on $i$?) can always be written in the form $$\frac{n^{p+1}}{p+1}+An^{p}+Bn^{p-1}+Cn^{p-2}+\cdots.$$ The solution states: The proof ...
1
vote
1answer
69 views

Solve for b: $a^2 = b^2 c^2 - b^4$

Given $c>b$ which leads to $b^2 c^2 - b^4 > 0$ is it possible to solve for b in the equation $$a^2=b^2 c^2 - b^4$$
2
votes
1answer
67 views

Compute $ax^3+by^3$ given $ax^2+by^2$ and $ax+by$.

Given are positive real numbers $A$ and $B$ and positive integers $a$ and $b$ such that $$ \begin{aligned} ax+by &= A\\ ax^2+by^2 &= B.\end{aligned} \tag{*}$$ What are the possible values of ...
0
votes
2answers
48 views

When finding the equation of a line.

When finding the equation of a line, how do you know whether to use the slope-intercept form or the point-slope form?
1
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3answers
33 views

Finding percentages by three weights

I have three groups with weights like Group A - 10 Group B - 5 Group C - 3 And I have a grand total of $100$ elements to assign to these groups based on their ...
1
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0answers
33 views

Resolve $ \dfrac{m^n}{(a^n-b^n)(a^{n+1}-b^{n+1})}$ where m = a $\times b$ into partial fractions

Find partial fraction of $ \dfrac{m^n}{(a^n-b^n)(a^{n+1}-b^{n+1})}$ where m = a $\times b$ $ \dfrac{m^n}{(a^n-b^n)(a^{n+1}-b^{n+1})}$ where m = a $\times b$ We can write it for partial fraction : ...
1
vote
2answers
281 views

Find minimum value of $27^{\sin x}+81^{\cos x}$

How to find the minimum value of the expression $$27^{\sin x}+81^{\cos x}$$
1
vote
1answer
80 views

If $a_n =\int^{\pi}_0 \frac{\sin(2n-1)x}{\sin x}dx$ ,

If $$a_n =\int^{\pi}_0 \frac{\sin(2n-1)x}{\sin x}dx$$ , then $$a_1,a_2,.....a_n$$ are in (a) A.P and H.P (b) A.P and G.P but not in H.P (c) G.P and H.P (d) A.P. ,G.P and H.P. I have ...
2
votes
2answers
144 views

Distributive Property Theory question

I've been wondering about the theory behind the distributive property lately. For example: 2(pi * r^2) is just 2 * pi * r^2. ...
5
votes
6answers
1k views

How to solve $\sin x \cdot\sin 2x\cdot\sin 3x + \cos x\cdot\cos 2x\cdot\cos 3x =1$

How to solve $\sin x \cdot\sin 2x\cdot\sin 3x + \cos x\cdot\cos 2x\cdot\cos 3x =1$ I don't know the solution for this. Help me! Thank all!
3
votes
3answers
3k views

Why is the derivative of sine the cosine in radians but not in degrees?

In radians the derivative of sine is the cosine. But why isn't this the same in degrees? According to this I'd first have to convert it to radians before this works. But when you look at the graph of ...
0
votes
2answers
149 views

What is power of number like (power of 2, power of 10)? and how to calculate power of number.

I know my question is very simple to somebody, but I'm still don't understand so far. And now my questions about this subject is: ...
1
vote
2answers
124 views

Calculating Needed Alloy Using Linear System Of Equations

I'm having troubles with this question which involves a linear system of equations. I keep encountering $x$ to be a negative number, which cannot be possible because you can't have a negative number ...