Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations and other symbolic-manipulation topics.

1
vote
3answers
44 views

How to find the smallest side of a triangle when the interior angles are unknown?

I am confused on how to find the answer for this problem. So far what I believe would apply is the triangle inequality but I'm not sure on how to use it. The figure $ABC$ is a triangle so as $BDC$. ...
1
vote
2answers
40 views

Sum of all real numbers $x$ such that $(\text{A quadratic})^\text{Another quadratic}=1$.

What is the sum of all real numbers $x$ such that $(x^2-5x+5)^{(x^2-7x+12)}=1$? So I know that $x^0=1$ and $1^x=1$. So, I can solve for them and find $x$, and add them up. Solving $x^2-7x+12=0$ ...
0
votes
0answers
13 views

Methods for finding range of a function

I'm aware that this can't be done in general, and that this is a broad topic, but are there some fast, applicable methods for finding range of a function ? Can we use intermediate value theorem, and ...
1
vote
2answers
46 views

Algebra problem asking for the sum of two people's ages.

A very simple problem tends to become very hard. Perhaps I am overthinking it. If the square of Winslow's age is added to Abby's age, the sum is 209. If the square of Abby's age is added to ...
1
vote
1answer
33 views

Summation formula for basic math question

I am reviewing some old retired case studies we have at work that we used to give to candidates interviewing. There is a short question on one part of the case that is pretty straight forward, but I ...
0
votes
2answers
40 views

Values of $x$ satisfying $\lfloor{x^2\rfloor}=\left(\lfloor{x\rfloor}\right)^2$

Prove that values of $x$ satisfying $\lfloor{x^2\rfloor}=\left(\lfloor{x\rfloor}\right)^2$ is $(0,\sqrt{2})\cup \mathbb{Z}$ My try: Its trivial that every integer satisfies the given equation. Now ...
-1
votes
0answers
15 views

$a_1 k^{b_1} + a_2 k^{b_2} + … + a_n k^{b_n} = 0$

Consider $a_1,a_2,...,a_n$ such that $a_1 + a_2 + ... + a_n = 0$. If there exists a real $k > 0$ different from $1$ and distinct real numbers $b_1,b_2,...,b_n$ such that $$a_1 k^{b_1} + a_2 k^{b_2}...
-2
votes
0answers
23 views

Dot and Cross Products [on hold]

A mechanic applies a force of 42 Newtons straight down to a ratchet that is 0.59 meters long. What is the magnitude of the torque when the handle makes a 38° angle above the horizontal?
0
votes
2answers
23 views

Express term a in formula in terms b, c, d etc

I have a formula that calculates $R$ with the following formula: $R = \dfrac{V\cos(α)\Big(V\sin(α) + \sqrt{V\sin(\alpha))^2 + 2gh}\Big)}{g}$ but in this case I need to know $V$ and I already know $...
0
votes
2answers
44 views

$x$ is a non-negative integer and $\sqrt{x^2 + \sqrt{x + 1}}$ is a positive integer.

Find non-negative integer $x$ such that $\sqrt{x^2 + \sqrt{x + 1}}$ is a positive integer. Because $\sqrt{x^2 + \sqrt{x + 1}} > x$, we let $x^2 + \sqrt{x + 1} = (x + y)^2, (y > 0)$ That means $...
1
vote
3answers
42 views

Solve in terms of $ln(5)$ and $ln(3)$: $15^{4y}=3^{4y+8}$

Here is my attempt, and I don't understand why it doesn't work (I apologise for my clunky MathJax): $(4y)ln15=(4y+8)ln3$ $(4y)(ln3+ln5)=(4y+8)ln3$ $4(y(ln3+ln5)=4(y+2ln3)$ $y(ln3+ln5)=(y+2ln3)$ $...
1
vote
1answer
25 views

Sliding scale range

I need to code a calculator and I'm struggling with the correct mathematical approach. It's to determine how much profit you can make given the hours and startup cash available. The example provided ...
5
votes
4answers
671 views

What is the Difference Between Formulating the Answer via Quadratic Formula and Factoring?

I'm quite eager to learn what is the difference between factoring quadratics (the $(x + a)(x + b)$ method), and using the typical formula (where $x = (-b \pm \sqrt{b^2 - 4ac})/2a$), and in what ...
0
votes
3answers
61 views

Solve for $x$ in $x^2-5x+2\sqrt{x^2-5x+3}= 12$

Solve for $x$ in $x^2-5x+2\sqrt{x^2-5x+3}= 12$ I've tried moving root term to one side and squaring both sides to get a $4$th-degree polynomial and find the roots that way. Is there any easier way of ...
1
vote
1answer
41 views

A non-abstract view of multiplying natural numbers

It is known that If you multiply more than 1 natural numbers together, it doesn't matter which order you put them in. If you multiply more than 2 natural numbers together, then it makes no difference ...
0
votes
2answers
39 views

Which simplified form of $-2\sqrt[3]{-250}$ is correct?

I simplified the cube root expression into two: $10\sqrt[3]{2}$ and $-10\sqrt[3]{-2}$. Both yield the same approximation when I solved them using calculator. Which is correct? Or both should be ...
0
votes
1answer
27 views

Solving equations over multiple lines: notational convention

Consider, for example, the following equation: \begin{align} x^2 - 2x = 1. \end{align} I'd like to solve the equation for $x$ in detail and am wondering how to write it down. Let me offer two ...
0
votes
2answers
37 views

Can I input negative angles into the cosine half-angle formula?

So the cosine half-angle formula says: Now, we know that co-terminal angles have equal cosines. Consider that $\cos (7\pi/4)$ = $\cos(-\pi/4)$. However, if you apply the half angle formula to $(7\pi/...
1
vote
0answers
22 views

Solving equation of form $a = \prod_i^N (c_i e^t + b)^{c_i}$

How can I go about solving the following equation for $t$? $$ \prod_{i=1}^4 p_i^{c_i} = \prod_{i=1}^4 \left(c_i e^{-t}+\frac14\right)^{c_i} $$ where $p_i \in (0,1)$, $c_i \in [-\frac14, \frac34]$, $\...
1
vote
2answers
50 views

All answers of $abcd=a+b+c+d-3$ in natural numbers

Given $$abcd=a+b+c+d-3$$ what are all possible 4-tuples $(a, b, c, d)$? I think that one answer lies within $(a,b,c,d)=(n,1,1,1)$ and all permutations of this answer? Is that right?
0
votes
1answer
25 views

Rational Equations: What are Excluded Values?

I have an example rational equation in a textbook which I was able to solve: $$\dfrac{3}{x-6} = \dfrac{5}{x}$$ Least common denominator is : $x(x-6)$ so: $$\frac{x(x-6)3}{x-6} = \dfrac{x(x-6)5}{...
2
votes
3answers
57 views

Equality of $2^x + 2^{4-x} \geq 8$

Want to find the value(s) of $x$ for which equality holds in $$2^x + 2^{4-x} \geq 8$$ I've found it by solving $2^x + 2^{4-x} = 8$: $$2^x + 2^{4-x} = 8 \Rightarrow 2^{2x} - 8 \cdot 2^x + 16 = 0 \...
1
vote
1answer
32 views

Find the parabola given two points and $y$-max (no axis of symmetry)

Given $(0,0)$ origin, point $(3,2)$ and $y\text{-max} = 5$, find the parabola. I tried to shift point $(3,2)$ down to $(3,0)$ so that it can become symmetric to origin. Then the vertex would be $(5,\...
0
votes
1answer
53 views

If x = .48 and y = -0.52, what is the formula for finding x(i) + y(i) = 0?

I'm not sure how to title this, so please feel free to edit title. I'm trying to zero out x and y, meaning how many x and y would I need for them to equal 0 -- i.e. be neutral? For example, if x = ...
0
votes
0answers
22 views

How do I test input values with imaginary numbers for rational inequalities?

The goal is to find all solutions (if any exist) for $$\frac{x^2+1}{y}\le0$$ Now, I set my constraints, {$y|y\ne0$}, and I set the expression in the numerator and denominator equal to zero to ...
0
votes
4answers
45 views

Error: $x^2 + 1 = 0$ has solution set $\{-1;1\}$

So, is it correct that the solution set of $x^2 + 1 = 0$ is [-1;1] ? Is the error in equation development or in the solution set? Help me, I don't know how to proceed in this question.
-1
votes
0answers
44 views

Need Formula… [on hold]

My customer tells me that my quoted rate is 100.9% higher than the rate he got from another competitor. What is the formula to get the rate that he received? There are hundreds of quotes I provided, ...
1
vote
1answer
31 views

What are the polar coordinates of $(2\sqrt3, 2)$?

My answer to this is $(4,\frac{π}6)$. But a calculator said that $(-4,\frac{7π}6)$ is also an answer, and there are infinitely many solutions. Is that correct?
2
votes
1answer
16 views

Find the appropriate value of $a$ s.t. $f(m,n)=g(m) h(n)$

For $f(m,n) = amn + 10m + 20n + 5$ is there a specific value for $a$ which enables us to write $f$ as product of a function of $m$ multiplied with a function of $n$, that is to write $f(m, n) = g(m) h(...
-1
votes
1answer
21 views

How to solve this non-linear equation [on hold]

This came from a regression: $$\text{Diameter} = 0.0531052 + 0.0443237 \cdot \exp (-0.0103633 \cdot \text{'Time elapsed'})$$ if diameter is $-0.052$ What will be time elapsed, can someone help me?
2
votes
1answer
77 views

Coefficients of $(1+x+x^2)^{2018}$

The question is How many of the coefficients of $(1+x+x^2)^{2018}$ are not divisible by 3? Somebody asked me the question, and I have no idea how to solve it. I am not sure if the coefficients are ...
0
votes
0answers
22 views

$k^x(z - y) + k^y(x - z) + k^z(y - x) \neq 0$

Let $k > 0, k \neq 1$, and $x,y,z$ distinct real numbers. Show that $$k^x(z - y) + k^y(x - z) +k^z(y - x) \neq 0.$$ My progress: Assume the expression is equal to 0. Write this as $$(z - y)k^x + ...
2
votes
3answers
28 views

How do I get from log F = log G + log m - log(1/M) - 2 log r to a solution withoug logs?

I've been self-studying from Stroud & Booth's excellent "Engineering Mathematics", and am currently on the "Algebra" section. I understand everything pretty well, except when it comes to the ...
0
votes
1answer
26 views

System of three non linear equations with three unknowns with random coefficients

I have the system of equations: $$ \begin{cases} Ax + By + Cz &= D \\ Exy + Fxz + Gyz &= H \\ Ixyz &= J \\ \end{cases} $$ Where $A,B,C,D,E,F,G,H,I,J$ are constant integers between 1 and 9. ...
0
votes
0answers
10 views

Is it possible to create an index relating two deviations but preserving the signal of one?

I have three temperatures I would like to compare. One is the body temperature ($T_{b}$), another is the reference ($T_{ref}$) and a third is the environment ($T_e$). I would like to create an index ...
-3
votes
1answer
24 views

continuity of function at one point [on hold]

We assume that the function $f(x)$ is defined by $$ f(x) = \left\{ \begin{array}{cc} x+1, & x\leq 0 \\ -\frac{1}{2}x + 7, & x > 0 \end{array} \right. $$ Find a real number such that the ...
0
votes
0answers
18 views

Newton's Generalization of the Binomial Theorem and big o notation

we know that : (Newton's Generalization of the Binomial Theorem) Let $x,y∈\mathbb{R}$ where $0≤∣x∣<∣y∣$ and let $α∈\mathbb{R}$. Then the expansion of the binomial $(x+y)^α$ is given by the ...
2
votes
2answers
54 views

(Baby maths) Sum of three positive integers = odd or even? + more info

I'm struggling to understand this basic question: $x$, $y$, and $z$ are positive integers. Is $x+y+z$ even? Supposedly you can derive the answer, from these two pieces of information separately: $x-...
0
votes
0answers
36 views

Is there a Singapore Math way to solve this with a bar model?

I have a series of word problems which involve finding two unknowns. In some problems, the relationship of the two unknowns is stated with a difference and a quotient, or a difference and a sum, or a ...
0
votes
0answers
18 views

Solve $S = \frac{P^K}{P^K + \left(1 - P^K\right) (1 - P)^K}$ for $K$

I have an inequality $$ S \leq \frac{P^K} { P^K + (1 - P^K) (1 - P)^K} $$ with $P \in [0..1]$ and $S \in [0..1]$. I need to solve it for $K$.
-3
votes
1answer
67 views

How do you write the sines of a binary expansion as an infinite series?

Consider the binary expansion: $$ \sum_{n=1}^\infty\frac{1}{2^n} $$ Now compare that to the sine of the above series: $$ \sum_{n=1}^\infty\sin \left(\frac{1}{2^n}\right) $$ The above is clearly ...
4
votes
3answers
49 views

How to find the angle in a protein which is inside of a triangle which appears inscribed in a circle?

I'm confused at which property or identity can be used to find the angle in a triangle when it looks inscribed in a circle but one of its sides doesn't appear to pass through the center. I'm also ...
1
vote
1answer
33 views

Best Buy marked a computer down from $\$4,500.00$ to $\$3,300.00$. What was the total percent markdown based on its original price?

I've tried doing this problem several ways and I can't seem to achieve the given answer of $41.3\%$.
-2
votes
1answer
44 views

Write a function for a river word problem

My problem: The river flows at 1.5 meters per second. Write a rational function rule for the total time required to travel a distance $d$ downstream and back in a river flowing at a rate $v_r$....
0
votes
0answers
22 views

Find the function given its entire graph, complete domain, full codomain and exact parts of its graph as other functions

I need to find a continuous function with its domain in the closed continuous interval $ [0-\frac{\pi}{2}] $. Its complete range or codomain is within closed continuous interval $ [1-\sqrt2] $. I also ...
2
votes
3answers
69 views

Nested trig functions (incl. inverse trig functions)

Edit: Although this problem has received a kind answer, I would still appreciate more comprehensive explanation. I am still rather confused. This problem has confused me a bit: The standard method ...
0
votes
1answer
26 views

Imaginary Numbers Case Work

Compute the number of ordered pairs of integers $(x,y)$ with $1\le x<y\le 100$ such that $i^x+i^y$ is a real number. So far I have found the cases in which they will work. But I am not sure how to ...
2
votes
0answers
35 views

reception area price

Question: The carpet chosen by Callis International for their reception area was $16.75 a square yard. What price should the store quote for the carpet if the reception area was 12 ft. by 6 in. by 8 ...
3
votes
3answers
52 views

Prove or disprove: If $A$ is $n\times n$ and $\exists\;m\in \Bbb{N}:\;A^m=I_n$, then $A$ is invertible.

Is this statement true? If $A$ is an $n\times n$ matrix and $A^m=I_n$ for some $m\in \Bbb{N}$, then $A$ is invertible. My trial Let $n\in \Bbb{N}$ be fixed. Then, $$[\det(A)]^m=\det(A^m)=I_n=1.$$ ...
0
votes
1answer
35 views

How can I find 2 unknowns when they are in the same equations?

So I have a question like this. Given that $x + 1/y = 3/2$ and $y + 1/x = 1/6$, what is the value of $x / y$? So basically I'm trying to get $x$ and $y$ on their own but I don't know how to because ...