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
7 views

How can one calculate the limit of (1/(x^2-9)) as x approaches -3 and 3 by hand?

Reviewing math for college after a gap year and so I know this is probably a pretty elementary question, but let me know if it has any interesting implications or alternative solutions or if it ...
0
votes
0answers
12 views

Within what angle does she need to throw her stone at to hit her opponents?

In curling, it is often necessary to hit and displace an opponent’s stone to win the end. Olivia would like to hit her opponent’s stone with her own stone. If she releases her stone at the hog line, ...
0
votes
2answers
50 views

Let $a$, $b$, and $c$ be positive real numbers.

Let $a$, $b$, and $c$ be positive real numbers. Prove that $$\sqrt{a^2 - ab + b^2} + \sqrt{a^2 - ac + c^2} \ge \sqrt{b^2 + bc + c^2}$$ Under what conditions does equality occur? That is, for what ...
1
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2answers
29 views

Why is Binomial Probability used here?

A test consists of 10 multiple choice questions with five choices for each question. As an experiment, you GUESS on each and every answer without even reading the questions. What is the ...
-1
votes
3answers
42 views

In most geometry courses, we learn that there's no such thing as “SSA Congruence”.

In most geometry courses, we learn that there's no such thing as "SSA Congruence". That is, if we have triangles $ABC$ and $DEF$ such that $AB = DE$, $BC = EF$, and $\angle A = \angle D$, then we ...
0
votes
1answer
17 views

Calculate weight of inputs in a simple equality

Suppose that we have this equality: 4+2+2 = 8 weight of 4 in this equality is 4/8=50%, ...
-2
votes
1answer
19 views

Find montly growth rate given principal and final value at year's end

If I start with £20,000 and want to end with a TOTAL of £500,000 after a 12 month period, how do I calculate the monthly growth rate?
-5
votes
3answers
31 views

If $f(x)=3x^3-2x^2+7x-5$ and $g(x)=5x^2+x-1$, what is $g(x) + f(x)$? [on hold]

If $f(x)=3x^3-2x^2+7x-5$ and $g(x)=5x^2+x-1$, what is $g(x) + f(x)$?
0
votes
0answers
21 views

Confused between cyclic sum and symmetric sums.

four variables $a, b, c, d$ are given, what is the symmetric and cyclic sum? I thought: $$\sum_{cyc} ab = ab + ac + ad + bc + bd + cd$$ And $$\sum_{sym} ab = 2(ab + ac + ad + bc + bc + ...
1
vote
3answers
46 views

$k2^x+2^x=8$, find the possible values of $k$

Find all the possible values of $k$ such that equation $$k2^x+2^x=8$$ has a single root. Find the root in the case. Can anyone give some hints for me? I have no idea how to solve it.
2
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2answers
59 views

If $\sin(x)=\sin(\pi/4 + x)$, then why isn't $x=x+\pi/4$?

I've been solving a question, If $\cos(x) + \sin(x)=\sqrt{2} \cos(\pi/2 - x)$ then find the value of $x$. We know that $\cos(x) + \sin(x)= \sqrt{2} \sin(\pi/4 + x)$. So, $$\sin(\pi/4 + x) = ...
0
votes
2answers
22 views

Straight line is tangent to the curve.

The straight line $y=mx+1$ is tangent to the curve $x^2+y^2-2x+4y=0$. Find the possible values of $m$. My attempt, Substitute the $y=mx+1$ into the equation $x^2+y^2-2x+4y=0$. ...
4
votes
0answers
24 views

Show that $p \in \left[\frac{4^m}{\sqrt{2m}},\frac{4^m}{\sqrt{2m+1}}\right]$

If the number of ways in which $m$ identical apples can be put in $2m$ boxes, so that no box contains more than one apple, is $p$, prove that $$p \in ...
2
votes
1answer
27 views

Intuitive Explanation For Why Dependent Equations Contain No Added Information?

I've always been taught that because dependent equations contain no added information they can be deleted without effecting the solution set. Now this makes sense to me if an equation is a constant ...
0
votes
0answers
18 views

Increment a number with a decrementing multiplier N times and have sum total always be equal to X?

(a working semi-related decrement function) 500 = 500 * 0.156048361 = 134.8206296 * 0.85 = 114.5975351 * 0.85 = 97.40790487 * 0.85 = 82.79671914 * 0.85 = 70.37721127 In this case T = 500, N ...
-1
votes
1answer
97 views

Solve the equation (very hard)

How to find all irrational solutions of the equation ...
0
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4answers
38 views

Do perfect square trinomials only have one root?

I apologize for the basic question, but I'm just now learning of perfect square trinomials in my math class. Google hasn't provided any relevant answers. Throughout all of the examples I have been ...
0
votes
1answer
31 views

Intersection of two lines in complex numbers given four points [on hold]

How to find the point of intersection of two lines, given four points, two of which are on each line, in complex numbers? Thank you!
0
votes
1answer
58 views

confusion about square root

I understand by convention ,if $\sqrt{x^2}=a$ , a is defined to be the positive root of x or the principal square root. but what does this mean for exponential equations- does $x^{0.5}=-5$ have no ...
0
votes
1answer
24 views

Probability for rolling an odd number and tossing a coin on heads

A coin is tossed and a die rolled. Find the probability of getting a head and an odd number. The answer is $\frac{1}{4}$. My reasoning is that rolling an odd number is $\frac{1}{2}$, and tossing a ...
1
vote
1answer
37 views

What is the difference between the slope and the angular coefficient?

What is the difference between the slope and the angular coefficient?
1
vote
1answer
30 views

Area of “Scalene” rectangles

My friend told me that from antiquity land revenue officials compute area of "reasonably" rectangular ( diagonal no matter) fields to assess tax taking the opposite sides average as .. $$ A = ...
-1
votes
1answer
28 views

Hearts Game Word Problem

Hearts is a game where the lowest score wins. We know this : The fourth player scored a $105$ The first three players scored a combined value of $103$ No scores are zero No score (except loser) can ...
0
votes
2answers
40 views

Find the height of the box.

A box contains 150 candles. It has a width that is five times the height of the box and a length that is thrice the width of the box. The volume of each candle is 1 $in^3$. Find the height of the box. ...
0
votes
3answers
49 views

Give a serious explanation of the difference between an equation and a function.

What's the difference between an equation and a function? I mean, I am not seeking for a high school-like answer like "an equation has an equals sign". I want to know what is the fundamental ...
-1
votes
1answer
32 views

How many different three-digit house numbers could be made?

a shopkeeper sells house numbers. she has a large supply of the numerals 4, 7 and 8, but no other numerals. how many different three-digit house numbers could be made using only the numerals in her ...
10
votes
3answers
1k views

Is there an algorithm to compute the degree of a polynomial?

Let $f\in k[X]$ be a polynomial in one unknown over any field (or any nice enough commutative ring, I imagine - it shouldn't matter) and suppose that all we can do to understand $f$ is to evaluate it ...
0
votes
2answers
28 views

How to calculate test score needed to maintain a certain average

I know the answer is probably very basic math, but I can't seem to figure it out. I want a 92 overall grade in math. -Test scores (make up 60% of grade): 86, 91, 90, 89 -Quiz scores (make up 25% of ...
-2
votes
1answer
41 views

The pascal triangle [on hold]

I really dont know what rules to apply to get this answer... but i know the following. I know that $$(a+b)^5$$ a decrease from $5$ to $0$ while $b$ increases .. eg \begin{array}{} ...
2
votes
0answers
33 views

Simplifying a recurrence relation.

I have the recurrence relation: $$g(k, 1, x) = k,$$ $$g(k, n, x) = \dfrac{1}{2} \log_{k}{\left(\dfrac{k^{g(k, n - 1, x)}x}{g(k, n - 1, x)}\right)},$$ and I would like to simplify it, if it is ...
-6
votes
5answers
48 views

Express $x$ algebraically with no nested radicals [on hold]

Given that $x= \sqrt{7+4\sqrt{3}}$, express $x$ algebraically with no nested radicals.
2
votes
4answers
117 views

How do I solve $2^x + x = n$ equation for $x$?

I need to solve the equation $$2^x + x = n$$ for $x$ through a programming-based method. Is this possible? If not, then what would be the most efficient way to approximate it?
0
votes
5answers
49 views

Points $A$, $B$, and $C$ are on the circumference of a circle with radius 2

Points $A$, $B$, and $C$ are on the circumference of a circle with radius $2$ such that $\angle BAC = 45^\circ$ and $\angle ACB = 60^\circ$. Find the area of $\triangle ABC$. I've drawn a circle ...
3
votes
2answers
37 views

Expressing a function in terms of compositions of three functions.

Express the function F in the form $f \circ g \circ h$. $$F(x)=\frac {9}{( x^2 + 7)}$$ I'm not sure how to get $x^2+7$ in the denominator. Here is what I tried: $$h(x) = (x+7)$$ $$g(x) = x$$ ...
1
vote
1answer
23 views

Difference quotient of $f(x)= 2-6x+4x^2$

I need to find $f(a), f(a + h)$, and the difference quotient $$\frac {f(a + h) − f(a)}{h},$$ where $h\neq 0$ and $f(x) = 2-6x+4x^2$. My work: $$f(a) = 2-6a+4a^2,\ \ f(a+h) = 2-6(a+h)+4(a+h)^2.$$ ...
-1
votes
3answers
46 views

Solve for $y$: $\frac{y+1}{y-1} = 10^{x^2}$ [on hold]

Can someone please show me the steps (all of them… yeah, even the obvious ones) to go from $$\begin{align}\frac{y+1}{y-1} = 10^{x^2}\end{align}$$ to ...
0
votes
0answers
28 views

Interpretation of Rates of Change [on hold]

Suppose $C(t)$ represents the number of cars currently at a particular station at time $t$. I am looking for possible interpretation of the following: (i) $C(t)\Delta t$ (ii)$\frac{C(t+\Delta t) - ...
14
votes
4answers
185 views

Product of cosines: $ \prod_{r=1}^{7} \cos \frac{r\pi}{15} $

Evaluate $$ \prod_{r=1}^{7} \cos {\dfrac{r\pi}{15}} $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\cos(A-B)] ...
4
votes
2answers
31 views

Number of ways to select subsets

In how many ways can two distinct subsets of the set $\text{A}$ of $k$ $(k \geq 3)$ elements be selected so that they have exactly two common elements? I started by choosing two elements (that ...
0
votes
1answer
34 views

What does it mean for a “formula to be undefined”?

I was covering the techniques used sketch rational functions of five different types as follows: However, then I encountered this: And, Ij just can't find out what it means for the formula to ...
6
votes
5answers
556 views

Systematically guessing integer roots of a cubic polynomial

Suppose I have a cubic equation, such as $$15x^3-4x^2-25x+14=0.$$ By the Hit and Trial method I know that one of the roots is $x=1,$ and hence I can solve the cubic equation with ease, as it will ...
-2
votes
0answers
19 views

Find the population [on hold]

Every year, the emigration rate from country A to B is 𝛼 (0 < 𝛼 < 1), whereas the emigration rate from country B to A is 𝛽 (0 < 𝛽 < 1). Note that the fluctuation in population of both ...
2
votes
3answers
327 views

Finding the roots of a different Quadratic equation from the roots of a Given Quadratic equation

The Question: If $\alpha$ and $\beta$ are the roots of the equation $ax^2+bx+c=0$... Then find the roots of the equation $ax^2-bx(x-1)+c(x-1)^2=0$ My Attempt: The new equation can be ...
4
votes
3answers
54 views

Square roots equations

I had to solve this problem: $$\sqrt{x} + \sqrt{x-36} = 2$$ So I rearranged the equation this way: $$\sqrt{x-36} = 2 - \sqrt{x}$$ Then I squared both sides to get: $$x-36 = 4 - 4\sqrt{x} + x$$ Then I ...
3
votes
1answer
34 views

Find the inverse of the function given:

$$f(x)= \frac{5x}{(x − 2)}$$ My work: $$y=\frac{5x}{(x-2)}$$ $$x=\frac{5y}{(y-2)}$$ $$x(y-2)=5y$$ $$xy-2x=5y$$ $$\frac{xy-2x}{5}=y$$ $$f(x)=\frac{(xy-2x)}{5}$$ Any help is appreciated.
0
votes
2answers
28 views

If the i-th, j-th, and k-th terms in an AP are in a GP with ratio r, find $r$ in terms of $i, j$, and $k$

If the i-th, j-th, and k-th terms in an arithmetic progression are in a geometric progression with ratio r, find r in terms of i, j, and k. This is my result: (1) if $ik \ne j^2$ then ...
2
votes
4answers
70 views

According to Stewart Calculus Early Transcendentals 5th Edition on page 140, in example 5, how does he simplify this problem?

In Stewart's Calculus: Early Transcendentals 5th Edition on page 140, in example 5, how does $$\lim\limits_{x \to \infty} \frac{\dfrac{1}{x}}{\dfrac{\sqrt{x^2 + 1} + x}{x}}$$ simplify to ...
0
votes
0answers
39 views

Can This Expression Be Simplified? (Involves Square Roots)

I started with the expression $$ \frac{4mlt(1-\sqrt{1-\frac{v^2}{c^2}})c^2}{\sqrt{1-\frac{v^2}{c^2}}} $$ and have ended up at: $$ \frac{4mlt(c^2 - c \sqrt{c^2-v^2})}{\sqrt{1-\frac{v^2}{c^2}}} $$ ...
1
vote
3answers
33 views

What is the unknown angle?

So first off I started with the pythagorean theorem to find the missing leg of the triangle. \begin{align*} 5^2 + b^2 ={}& 8^2 \\ 25 + b^2 ={}& 64 \\ 64 - 25 ={}& 39 \\ \text{missing ...
1
vote
2answers
35 views

Representation in the complex plane.

Determine the set of representations in the complex plane for which: (a) $\frac{z-1}{z+1}$ is a real number; (b) $\frac{z-1}{z+1}$ is a pure imaginary number.