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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3
votes
2answers
65 views

Find the value of $ab+ 2cb+\sqrt3 ac$?

Three positive real numbers $a,b,c$ satisfy the equations $a^2+\sqrt3 ab+b^2=25$, $b^2+c^2=9$ and $a^2+ac+c^2=16$ .Then find the value of $ab+ 2cb+\sqrt3 ac$? Is there some way to find the desired ...
1
vote
2answers
41 views

How to solve $\frac{1}{(|x| - 3)}$ $\lt$ $\frac{1}{2}$?

$\frac{1}{(|x| - 3)}$ $\lt$ $\frac{1}{2}$ $x$ can be $\ge$ 0 or $\le$ 0 Case 1 :- $x$ $\ge$ 0 $therefore$, $\frac{1}{(x - 3)}$ $\lt$ $\frac{1}{2}$ $\Rightarrow$ $\frac{1}{(x - 3)}$ - ...
1
vote
1answer
20 views

Is this solution for a broadwalk problem correct?

Question: Distance, Speed, and Time A boardwalk is parallel to and 210 ft inland from a straight shoreline. A sandy beach lies between the boardwalk and the shoreline. A man is standing on ...
1
vote
0answers
46 views

Defining Logic Algebraically, Math Functions & Integers

Introduction I wanted to define some functions algebraically to be used as "logical conditions" that would be assigned to a term $t$ to "control" its value. Or in some other words, I wanted to ...
2
votes
1answer
66 views

Solve given equation $4^{(x-2)(x+3)} - 64^{(x-3)} = 0?$

Solve given equation $4^{(x-2)(x+3)} - 64^{(x-3)} = 0?$ My attempt: I've attempted to solve this question, but isn't it impossible to solve, i.e has already been simplified completely? ...
3
votes
2answers
53 views

Finding Roots of tenth degree polynomial

I know that there are no explicit formulas to find roots for polynomials of degree higher than $4$. I have to find all the roots of the polynomial $ f(z) = 1+z^2+z^4+z^6+z^8+z^{10}$ I found two ...
0
votes
2answers
63 views

Wolfram answer is different for the integral $\sqrt{\frac{x}{2-x}}dx$

$$I=\sqrt{\frac{x}{2-x}}dx=\int \frac{xdx}{\sqrt{2x-x^2}}=\frac{-1}{2} \times \int\frac{(2-2x-2)dx}{\sqrt{2x-x^2}}$$ so $$I=\frac{-1}{2}\int\frac{(2-2x)dx}{\sqrt{2x-x^2}}+\int ...
3
votes
3answers
83 views

Integrating $\int \frac{\sqrt{x^2-x+1}}{x^2}dx$

Evaluate $$I=\int\frac{\sqrt{x^2-x+1}}{x^2}dx$$ I first Rationalized the numerator and got as $$I=\int\frac{(x^2-x+1)dx}{x^2\sqrt{x^2-x+1}}$$ and splitting we get ...
1
vote
1answer
27 views

Evaluate a difference quotient - Pre Calculus Homework

Evaluate the difference quotient: $f(x)=x^2-x+1$, $\displaystyle \frac{f(2+h)-f(2)}{h}$, $h\not=0$ I have not been able to solve this problem the farthest I have gotten to is $\displaystyle ...
3
votes
7answers
141 views

Rational Expression equivalent form

EDIT: I know how to find the answer, but does anyone know why plugging in numbers for x does not work? The Question: If the rational expression $\frac {3x^2}{3x-1}$ is rewritten in the equivalent ...
-4
votes
1answer
56 views

Two pipes cement truck time equation

Two pipes are used to pump a cement mixture out of a truck. The pipes work at the same rate. When pipe A works alone it can empty the truck in 45 minutes. When both pipes are used together they ...
0
votes
2answers
28 views

Substitute for y'

The problem was: $$ x^x=\mathrm{e}^{x-y} $$ I was able to solve it to(By Implicit differentiation) : $$ x^x\left(\ln\left(x\right)+1\right)=\mathrm{e}^{x-y}\left(1-y'\right) $$ But how do I ...
-1
votes
2answers
17 views

Co-Ordinates and midpoint [closed]

I can find the markscheme for the following questions- but they do not show the working so I am confused on how they managed to get the answers? If you would please be able to help me that would be ...
1
vote
1answer
17 views

Finding the equation for a line tangent to a parametric curve

I have the parametric equation $x = 2t - 1$ $y = 3t + 5$ $t = -1$ (defined as $t_0$) I am trying to find the line tangent to it. My book says if $x'(t_0) \not = 0$ then you can use the ...
2
votes
2answers
46 views

How do I simplify $\frac{1}{1+\frac{x^2}{2}+\frac{5x^4}{48}+\frac{7x^6}{576}\dots}$ using long division?

The infinite series $\frac{1}{1+\frac{x^2}{2}+\frac{5x^4}{48}+\frac{7x^6}{576}\dots}$ is supposed to simplify to $1-\frac{x^2}{2}+\frac{7x^4}{48}+\frac{19x^6}{576}\cdots$ but I don't know how this was ...
6
votes
3answers
84 views

Polynomial equation: $P(\sin t) = P(\cos t)$

Let $P(X)$ be a polynomial with real coefficients such that $P(\sin t) = P(\cos t), \, \forall t \in \mathbb R$. Prove that there exists a unique polynomial $Q(Y)$ with real coefficients, such that ...
1
vote
1answer
31 views

How to prove by induction that $\frac{a^n+b^n}{2}\geq\left(\frac{a+b}{2}\right)^n$?

I'm about to prove that for any $a,b>0$ and $n\in\mathbb{N},$ the inequality: $\frac{a^n+b^n}{2}\geq\left(\frac{a+b}{2}\right)^n$ holds. By induction I get: ...
-3
votes
1answer
58 views

How to resolve this fractional equation? [closed]

Can anyone help and explain how can i resolve this equation? $$1650=\frac{1}{\frac{1}{x_1}+\frac{1}{x_2}}$$
4
votes
0answers
90 views

Given $a+b+..=a^7+b^7+..=0$ show that $a(a+b)..=0$

Question: Suppose $a,b,c,d$ are real numbers such that $a+b+c+d=a^7+b^7+c^7+d^7=0$ Show that $a(a+b)(a+c)(a+d)=0$ My attempt: Using $a+b+c+d=0$, I get $a(a+b)(a+c)(a+d)= 0 \implies a=0, ...
3
votes
2answers
83 views

Solve the equation $x^3-6x-6=0$

Evaluate the roots of $$x^3-6x-6=0$$ I solved it using Cardano's method, but I'm looking for other elementary approaches through substitutions and properties of polynomials. ...
1
vote
5answers
43 views

Is the following solution correct?

Question: $ \sqrt{x^2 + 1} + \frac{8}{\sqrt{x^2 + 1}} = \sqrt{x^2 + 9}$ My solution: $(x^2 + 1) + 8 = \sqrt{x^2 + 9} \sqrt{x^2 + 1}$ $=> (x^2 + 9) = \sqrt{x^2 + 9} \sqrt{x^2 + 1}$ $=> (x^2 + ...
9
votes
2answers
100 views

Fundamental Theorem of Algebra for highschool

My teacher has told me about the Fundamental Theorem of Algebra, but I can't seem to find any proofs on it which I can understand. For something so important I'm hoping to find a proof that a ...
1
vote
3answers
97 views

How to solve $x^3 = 1$?

My intuitive side tells me to take the cube root of both the sides and get the answer $1$. However, I realize that it might be a problem for I'll lose solutions as given here: Is it the case that ...
0
votes
0answers
14 views

Continuous compounding for multiple interest rates

If I have a quantity $A$ and grow it over one time period at an interest rate $r$, then I can find an equivalent continuous compounding rate $r'$ such that $A\exp(r')=A(1+r)$, or $r'=\ln(1+r)$. What ...
3
votes
2answers
35 views

Find $a, b, c$ values from function $y=ax^2-bx+c$ and minimum value $D$

The problem reads like this: The quadratic function which takes the value $41$ at $x = -2$ and $20$ at $x = 5$, is: $y = Ax^2-Bx+C$ The minimum value for this function is: $D$ I order it ...
0
votes
3answers
37 views

How to perform long division on polynomials.

$$\frac{ 9x^6 - 4x^5 + 0 + 0 + 3x^2 + 0 - 1 }{ x^2 - 2x + 1 }$$ I know it starts with $\ { 9x^4 }$but the next two numbers are $\ { 14x^3 }$ $\ { 19x^2 }$. How do you come up with those two numbers? ...
0
votes
1answer
24 views

Absolute value inequality explanation

I was solving the inequality $-4 \le \left|\frac {x+4} {2-x} \right| \le 4$ and I first wrote the domain, which is $(-\infty,-4] \cup (-4,2) \cup (2, \infty)$ and I got the solution that $x \le \frac ...
0
votes
4answers
66 views

How to square both the sides of an equation?

Question: $x^2 \sqrt{(x + 3)} = (x + 3)^{3/2}$ My solution: $x^4 (x + 3) = (x + 3)^3$ $=> (x + 3)^2 = x^4$ $=> (x + 3) = x^2$ $=> x^2 -x - 3 = 0$ $=> x = (1 \pm \sqrt{1 + 12})/2$ I ...
1
vote
2answers
29 views

What is the solution of $\sin z=\cosh 4$?

What is the solution of $\sin z=\cosh 4$? By putting $z=x+iy$ I managed to find that the real part of $z$ is $x= \frac \pi 2+2n\pi $, but the imaginary part is contradictory giving negative value of ...
0
votes
1answer
37 views

What's wrong with my solution for equation $x^{1/2} + 3x^{-1/2} -10x^{-3/2}$?

The solution given in the website: Here's my solution: What am I doing wrong?
0
votes
1answer
23 views

Solve for $s$: $s - \frac{s}{ 2^{\frac{f}{12}}} = w$

I have a (probably) relatively simple algebra question. The values $w$ & $f$ are unknown constants. Solve for $s$: $$s - \frac{s}{ 2^{\frac{f}{12}}} = w$$
0
votes
3answers
72 views

Why doesn't $x^4 = -16$ have a solution?

The bottom of the page: Solution to a problem. Shouldn't the answer be $x=-2$?
0
votes
1answer
30 views

Expressing constant of integration as its natural log in 1st order linear ODE

What is the complete reasoning behind constants of integration, specifically in the case of the natural log? Given $\int\left(\frac{1}{250-x}\right)dx=\int(dt)$, I obtain: ...
-2
votes
1answer
20 views

Trigonometric Identity Similar to Tangent Addition Identity

I found the following trigonometric relationship in a paper and I can't seem to derive it. $1/2 arctan(\frac{2\sqrt{l}}{l-1}) = arctan(\sqrt{l})$ Where $l$ is a natural number. It seems similar ...
3
votes
3answers
62 views

What are the solutions of $|x+y|=|x|+|y|$?

So I am having a problem in solving this type of equation. The problem I am dealing with is... $$\left|(2x-1) + \frac{3x-1}x\right| = \left|2x-1\right| + \left|\frac{3x-1}x\right|$$ Please help me ...
2
votes
2answers
20 views

Having trouble isolating for y

I have this equation $x^3(y-1)^2 = 4y^4(x-1)^3$ and I'm trying to isolate for y. No matter what I do, I keep getting $\frac{y-1}{y^2} = \frac{2(x-1)^{3/2}}{x^{3/2}}$ I have no idea how to isolate ...
-1
votes
2answers
49 views

How to perform long division on polynomials? [closed]

$$\frac{x^6 - 3x^5 + x^4 - 2x^3 - 3x^2 + x - 3}{x^2 + 1}$$ I got the answer in the book, but I can't figure out how it comes up with the x for $$\ { -2x^3 + 3x^3 }$$
0
votes
1answer
30 views

Find the smallest number of toys that person had

A person had a number of toys to distribute among children . At first he gave $2$ toys to each child , then $4$ , then $5$ ,and then $6$ , but was always left with one . But if he had given $7$ toys ...
1
vote
2answers
32 views

Tangents to an ellipse passing through an external point

I came across the following problem from a Calculus course. Given the equation of an ellipse $9x^2+4y^2=36$ and a point $P(4,0)$. Find the equations of the two tangents to the ellipse, passing through ...
0
votes
2answers
59 views

Is it the right definition of a median $\frac{1}{2} \sum_{i=1}^n x_i$

Might be silly question, but is it a right formal way to define a median? $\frac{1}{2} \sum_{i=1}^n x_i$ Thanks
0
votes
1answer
51 views

How to solve $a^x + bx = c$?

Is it possible to solve equation in form $$a^x + bx = c$$ algebraically, where $a$, $b$ and $c$ are given, $a, b, c \in \Bbb{N}$ and $x$ is unknown? If it's solvable algebraically, how would you ...
3
votes
3answers
78 views

Real roots of $x^6+15x^2-60x+1$

How can I prove that $f(x)= x^6+15x^2-60x+1=0$ can't have three real roots. First I derive two times for see the signs of derivatives and got that $ 6x^5+30x-60=0$ and $ 30x^4+30=0$ which is always ...
4
votes
1answer
35 views

An operation on a whole is equal to an operation on each part?

I've been pondering this question as to how and when you can perform an operation on a complete "unit" and the answer is the same when performing the operation on the individual parts of the "unit" ...
18
votes
3answers
12k views

If there are $74$ heads and $196$ legs, how many horses and humans are there? [closed]

I was going through some problems then I arrived at this question which I couldn't solve. Does anyone know the answer to this question? One day, a person went to a horse racing area. Instead of ...
0
votes
1answer
23 views

Tangent makes with the x-axis

A curve has equation $y=\frac{4}{3x-4}$ and $P$(2,2) is a point on the curve. Find the angle that this tangent makes with the x-axis. Can anyone explain this ?
1
vote
1answer
22 views

Constant angles and powers

One can verify without difficulty that for all triple $(a,b,c)$ of real numbers greater than $1$, with $a\le b\le c$, and for all positive integer $n$, the equality $$a^n+b^n=c^n\qquad (*)$$ ►it has ...
1
vote
1answer
24 views

How to calculate percentage of value in arbitrary range

this is similar to a previous question so I've pasted the original question here with my amendment. "I have a slider that returns a value in a given range, so: min: 174 max: 424 slider current ...
1
vote
2answers
29 views

Help me prove a modular congruence!

Show that $a^{42} \equiv 1 \pmod{1764}$ if $\gcd (a, 1764) = 1$. Use Euler's theorem. Hint: $1764 = 4 \cdot 9 \cdot 49$ Hint: if t is a common multiple of $\phi(m)$ and $\phi(n)$, where ...
1
vote
2answers
37 views

For a polynomial $p(x)$ of odd degree, how many real roots does $(p(x))^2$ have?

I know that an odd degree polynomial $p(x)$ has at least one real root; what about $(p(x))^2$? It is even, so can it not possess any real roots or have two real roots? Also, let $p(x)$ be an odd ...
10
votes
2answers
1k views

Solve an equation of 4th degree

Solve the following equation: $$x^2+\dfrac{81x^2}{(9+x)^2}=40.$$ Unfortunately I have no ideas because after expanding I get an equation of 4 degree.