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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 ...

0
votes
4answers
41 views

How to solve $6x^{2/3}-12-2x=0$?

How to solve $6x^{2/3}-12-2x=0$? I need it to prove that the graphic of $y=6x^{2/3}$ never meets the graphic of function $y=12+2x$.
0
votes
3answers
30 views

Finding $(\alpha - \gamma)(\alpha - \delta)$ if they are roots of given quadratic equations

If $\alpha, \beta$ are roots of the equation $x^2 + px - q = 0$. $\gamma , \delta$ are roots of equation $x^2 + px -r$, then find the value of $(\alpha - \gamma )(\alpha - \delta)$. Answer - $q-r$ ...
-2
votes
2answers
32 views

Find values of $a$

How do I find an integer $a$ such that that : $$ (x-a)(x-12)+2$$ can be factorized as $(x+b)(x+c)$ where $b$ and $c$ are integers too. I have tried expanding the equation and taking the ...
0
votes
3answers
28 views

Finding sides of rectangle

It is given that area of rectangle = its perimeter and the diagonal is 3√5. We are to find it's area by use of quadratic equations. The problem is that I'm getting a 4 degree equation when I ...
-1
votes
0answers
28 views

Parabola-GRE doubt

How can one interpret just by seeing the equation of parabola that it opens UP/DOWN and or centered around $x$- or $y$-axis. I know parabola equation as $ax^2+bx+c$, I know that it opens up when $a&...
0
votes
1answer
43 views

Given that cot(m)=0.75 and cos(m)<0, what is the value of sin(m)?

One of my homework problems is "Given that $\cot(m)=0.75$ and $\cos(m)<0$, what is the value of $\sin(m)$?" I keep getting $\sin m=\frac{-4}{5}=-.8$ which isn't an option My options are: A- $\ -....
0
votes
3answers
58 views

What is the mathematical property that dictates that $f(x+y) = \tan(x+y)$ given that $f(x) = \tan(x)$?

We were asked in our Calculus class to prove that, $f(x+y) - f(x) = \frac {\sec^2(x) \tan(y)} {1 - \tan(x) \tan(y)}$ given that $f(x) = \tan(x)$ I have gotten so far as: $$f(x+y) - f(x)$$ $$\tan(...
1
vote
1answer
40 views

Find number of zeros at the end of $2007!$ using greatest integer function?

How to solve this question using greatest integer function? I used logical thinking to get answer but I want to solve this question using greatest integer function. Answer: $500$ Please explain ...
2
votes
3answers
36 views

Complex number question - spurious solutions…

$z_1 = 2 + 3i$ and $z_2 = 3 - 4i$ The complex number $z = x + iy$ is such that $\frac{z + z_1}{2z - z_2} = 1$. Find the value of $x$ and the value of $y$. Method 1: $$z + z_1 = 2z - z_2 ...
0
votes
1answer
22 views

Probability Question dice [on hold]

for the first part, I think its £10.50? Can anyone help me with the second part? thank you.
2
votes
1answer
20 views

Permutation question involving seating

In a class of 20 students having 20 chairs in 5 rows of 4 each. If the class has 10 boys and 10 girls, in how many ways, can the student's be placed in the chair's such that no boy is sitting in ...
0
votes
2answers
36 views

In a crowd I count $100$ heads, $162$ eyes, and $283$ legs. [on hold]

Can anyone help me with this ? I seem to think its 54?
0
votes
1answer
17 views

For an operator $\hat A$, if $\langle \hat A\rangle=0$, does this mean that $\langle \hat A\rangle\langle\hat A\rangle=0\,?$

Consider an arbitrary operator $\hat A$, let's suppose that $\langle \hat A\rangle=0$. Does this mean that I conclude that $$\langle \hat A^2\rangle=0\,?$$ I used to think that the answer to this ...
-1
votes
2answers
38 views

How to prove $x^4+y^4\leq (x^2+y^2)^2$?

I want to prove: $$x^4+y^4\leq (x^2+y^2)^2 \tag 1$$ We always have $x^2\geq 0$ and $y^2\geq 0$. Therefore we also have $x^4\geq 0$ and $y^4\geq 0$. Addition of the inequalities gives $$ x^4+y^4\geq ...
-1
votes
2answers
30 views

How to solve “If $\tan x = 24/7$ and $x$ is in the first quadrant, then find $\cos(2x)$.”

One of my problems for homework is "If $\tan(x)=24/7$ and $x$ is in the first quadrant then find $\cos(2x)$." I keep getting to $\cos(7/25 + 7/25)$ which evaluates to $.84726$, which isn't an option. ...
2
votes
2answers
43 views

Roots of unity and large expression

Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find $$\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 + \...
0
votes
0answers
25 views

Ratio of triangle to that formed by angular bisector.

If the bisectors of the angles of triangle $ABC$ meet the opposite sides in $A',B', C'$ prove that ratios of the area of the triangles $A'B'C'$ and $ABC$ is $$2\sin\dfrac{A}{2}\sin\dfrac{B}{2}\sin\...
1
vote
3answers
53 views

Prove $\forall x,y \in \mathbb{R} :\lfloor{x+y}\rfloor=\lfloor{x}\rfloor+\lfloor{y}\rfloor∨\lfloor{x+y}\rfloor=\lfloor{x}\rfloor+\lfloor{y}\rfloor+1$

Prove $∀x,y\ (x,y\in \mathbb{R}: \lfloor{x+y}\rfloor=\lfloor{x}\rfloor+\lfloor{y}\rfloor∨\lfloor{x+y}\rfloor=\lfloor{x}\rfloor+\lfloor{y}\rfloor+1)$ So, I let $\lfloor{x}\rfloor=m ≡ m≤x<m+1$ $...
0
votes
1answer
22 views

Using the exponential growth and decay formula for compound interest

I have what seems to be a rather simple question but one that is confusing me a lot. When looking at standard exponential growth/decay models(such as the decay of Carbon-14 etc), we can use the ...
1
vote
1answer
38 views

Could someone help clarify the meaning of this pre-calculus question on Binomial Expansion?

I'm taking an online pre-calculus course and I'm currently working on a unit about Binomial expansion. I've come across an oddly worded (at least to me) question on coefficients: "What coefficient ...
-4
votes
2answers
41 views

Basic identity Question [on hold]

How do I go about solving this as an identity question? How do i make the left side the same as the right side?: $$\frac{x^3-1}{x-1}=x^2+x+1 $$
0
votes
0answers
39 views

If sin(ax) = z*sin(x), is it possible to isolate x?

If i have the equation of the form $$sin(ax) = zsin(x)$$, where $a$ is a whole number, and $x$ is any real number, how do I isolate the variable $x$? Is it even possible? There isn't an easy way to ...
1
vote
2answers
38 views

Reduction Formula for $I_n=\int \frac{dx}{(a+b \cos x)^n}$

Reduction Formula for $$I_n=\int \frac{dx}{(a+b \cos x)^n}$$ I considered $$I_{n-1}=\int \frac{(a+b \cos x)dx}{(a+b \cos x)^n}=aI_n+b\int \frac{\cos x\:dx}{(a+b\cos x)^n}$$ Let $$J_n=\int \frac{\cos ...
11
votes
7answers
652 views

Distance between two stations

A railway line is divided into $10$ sections by the stations $A, B, C, D, E, F, G, H, I, J$ and $K$. The distance between $A$ and $K$ is $56$ km. A trip along two successive sections never exceeds $12$...
0
votes
1answer
45 views

What are the values of $z$ and $y$?

So we all know that $(x + a)(x + b) = x^2 + x(a + b) + ab$ Now I’m stuck to turn this: $x^2 - 30x - 56$ into something like $(x+z)(x+y)$. Here’s what I’ve done: We know $y+z= -30$ and $yz = -56$ and ...
0
votes
3answers
38 views

Roots of cubic equation with integer coefficients

Consider a cubic equation with integer coefficients (coefficient of $x^3$ being $1$). In hit and trial method, we assume each divisors of the last term as the root and check if it satisfies the ...
0
votes
0answers
48 views

Explanation of the proof as given in Elements of Algebra by Euler [on hold]

When we have removed fractions from a equation for the third digree, according to the manner which has been explained, and none of the divisors of the last term are found to be a root of the equation, ...
0
votes
3answers
33 views

If $\sin q\ne \cos q$ and $x,y,z$ satisfy the equations $x\cos p-y \sin p+z=\cos q+1$, $x\sin p+y\cos p+z=1-\sin q$, $x\cos(p+q)-y\sin(p+q)+z=2$

If $\sin q\ne \cos q$ and $x,y,z$ satisfy the equations $$ x\cos p-y \sin p+z=\cos q+1\\ x\sin p+y\cos p+z=1-\sin q\\ x\cos(p+q)-y\sin(p+q)+z=2 $$ then find the value of $x^2+y^2+z^2$ I multiplied ...
11
votes
9answers
3k views

Does there exist a right triangle with area 7 and perimeter 12?

This question is really trivial. I can prove that there is no right triangle with area 7 and perimeter 12, but what I do is solve the following system: if $a$, $b$ and $c$ are, respectively, the two ...
0
votes
1answer
134 views

Need help solving for $x$ the equation $x^2 - x -1 = 0 $ [on hold]

I need help solving for $x$ the equation $x^2 - x -1 = 0$. I actually already have the answer. It's $$\frac{1\pm\sqrt5}2\,.$$ However I cannot figure out the steps to get to the answer. I thought $x^...
-5
votes
0answers
42 views

I did this math, but my answer is incorrect. What mistake did I make? [on hold]

I'm stating my attempt below:- Let the loan of Zami is $x$ and the loan of Simi is $y$. Now in case of Zami, $I=xnr$ $\implies x=\frac{I}{nr}$ $\implies x=\frac{I}{\frac{2*10}{100}}$ $\implies x=\...
-1
votes
1answer
62 views

Best math workbooks?

I'm learning math with Khan Academy, and currently I'm doing polynomials (division, factoring, quadratics etc.), and I want to do more exercises than it's available there. Do you know any good ...
0
votes
2answers
33 views

Value of c in equation $1+\log_2(2x^2+2x+\frac{7}{2}) \ge \log_2(cx^2+c)$ for at-least one solution

Find all the values of the parameters c for which the following inequality has at least one solution. $1+\log_2(2x^2+2x+\frac{7}{2}) \ge \log_2(cx^2+c)$ I will elaborate the following steps 1.$\...
9
votes
2answers
132 views

Sum of fifth roots of roots of cubic.

$a,b,c$ are the (real) roots of $x^3-16x^2-57x+1=0$. Prove that $\sqrt[5]{a} + \sqrt[5]{b} +\sqrt[5]{c} = 1 $ ................................................................................ edit : ...
3
votes
2answers
60 views

Probability of choosing two marbles with the same color is equal to the probability of choosing two marbles with different colors

Problem statement: John and Daisy have devised a system for walking the dog. In a bag they have red marbles and blue marbles. If two marbles of the same color are picked from the bag, then John walks ...
2
votes
2answers
44 views

Solve the following problem, $u'(t)+p(t)u(t)=0,\;\;u(0)=0,$ $p(t)=\begin{cases}2& 0\leq t< 1,\\1 &t\geq 1\end{cases}.$

Using Laplace transform, solve the following problem. $$u'(t)+p(t)u(t)=0,\;\;u(0)=0,$$ $$p(t)=\begin{cases}2& 0\leq t< 1,\\1 &t\geq 1\end{cases}.$$ Here's what I've done: Taking the ...
3
votes
2answers
64 views

$x,y,z>0$, $x+y+z=1$. Prove $x^2+y^2+z^2+3xyz \geq \frac{4}{9}$

Let $x$, $y$, $z$ be positive real numbers such that $x+y+z=1$, then $$ x^2+y^2+z^2+3xyz \geq \frac{4}{9}.$$ From Cauchy–Schwarz inequality, we have $$ \left( x^2+y^2+z^2 \right)\left(1^2+1^2+1^2\...
1
vote
1answer
38 views

Looking for Teacher-Reviewed books [on hold]

I am a fairly new math teacher, and will be teaching high school Precalculus this year! I am lucky, because I get to choose the textbook for my class! However, I am frustrated because the previous ...
-3
votes
1answer
43 views

Find the value(s) of the constant $k$ such that the system of linear equations: [on hold]

Find the value(s) of the constant $k$ such that the system of linear equations: \begin{align} 9x + ky &= 3,\\ kx + y &= 1 \end{align} has (i) No solution. (ii) An infinite number of ...
-4
votes
6answers
95 views

finding the zeros of a quadratic function [on hold]

I am learning quadratic equations and got stuck at an exercise which asks to find the minimum point of this parabola: The solution starts off by stating that: The parabola has zeros at $x = -2.2$ ...
-1
votes
1answer
58 views

If $a, b, c$ are positive real numbers such that $abc=1$ prove that $\frac{a^3}{(a-b)(a-c)} + \frac{b^3}{(b-a)(b-c)} + \frac{c^3}{(c-b)(c-a)} ≥ 3$ [duplicate]

If $a, b, c$ are distinct positive real numbers such that $abc=1$, prove that $$\frac{a^3}{(a-b)(a-c)} + \frac{b^3}{(b-a)(b-c)} + \frac{c^3}{(c-b)(c-a)} ≥ 3.$$ I tried to do this problem by ...
0
votes
4answers
61 views

If $x:y=2:1$ and $y:z=2:1$, then $x$,$y$,$z$ are continued proportional? and/or $z:x=1:4$? and/or $y^2+z x=4yz$?

I will prove that I attempted this math by listing the things that I know. $y=2z$ $x=2z^2$ $x=yz$ Also, did you notice something? If $x=8, y=4,z=2$ then everything works out. I figured this while ...
0
votes
3answers
39 views

Factorising cubic polynomial given 1 factor

Given that $x^3-x^2-17x-15 = (x+3)(x^2+bx+c)$ where $b$ and $c$ are constants, find the values of $b$ and $c$. I don't know how to easily solve this question. I could use polynomial long division but ...
0
votes
3answers
38 views

It is it possible to always find arbitrarily large solutions to this equation?

I have been playing around with Collatz sequences a lot recently, and so now I want something slightly different but not too different, if you get what I mean. Solving puzzles online doesn't quite ...
-3
votes
0answers
36 views

Why sin(1 degree) couldn't have an exact mathematical value?

Some of cubic equations may have answers in radical form and exact value but the one related to sin(1 degree) couldn't be solved,Why
0
votes
0answers
43 views

Solve for θ: $y = \arctan\Bigl(\frac{x· \sin(θ)- \sin(θ·x)}{x·\cos(θ)-\cos(θ·x)}\Bigr)$ [duplicate]

I'm stuck here: $$\tan(y)=\frac{x·\sin(θ)-\sin(θ·x)}{x·\cos(θ)-\cos(θ·x)}$$
-2
votes
0answers
20 views

Finance simple intrest [on hold]

The difference between 4 year simple and compound interest is 7630.19 what is principal amount
0
votes
4answers
74 views

Proving $\frac{1}{\sqrt{x}}\ge \frac{2}{x+1}$ for $x> 0$ [on hold]

Prove: $$\frac{1}{\sqrt{x}}\ge \frac{2}{x+1}, \quad\forall x>0$$ Yeah, pretty much it. I've tried all manner of rearranging and just can't seem to get it. Thanks.
2
votes
4answers
50 views

Find the values for $a,b,c,d$

Given $$x^3+4x^2y+axy^2+3xy-bx^cy+7xy^2+dxy+y^2=x^3+y^2$$ for any real number $x$ and $y$ , find the value of $a,b,c,d$
0
votes
1answer
48 views

If $a+b+ab=80$, $b+c+bc=120$ and $c+a+ac=24$. Then $ab+bc+ca=?$

If $a+b+ab=80$, $b+c+bc=120$ and $c+a+ac=24$ then what is $ab+bc+ca$? I tried substitution to obtain $b$, but then I realized it'd be a quadratic equation which has two values. Likewise, $a$ and $c$ ...