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Mathematics 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.

This tag is for questions typically taught in precalculus, as well as elementary algebra.

These topics include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomials, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

47234 questions
5
votes
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Area of $\triangle ABC$ whose sides $a,b,c$ satisfy $0\leq a \leq1;1\leq b \leq2;2\leq c \leq3$ is

The maximum area of a triangle whose sides $a,b,c$ satisfy $0\leq a \leq1;1\leq b \leq2;2\leq c \leq3$ is $\bf{My\; Try::}$ Area of $\displaystyle \triangle ABC = \frac{1}{2}ab\sin C = \frac{1}{2}ab\cdot \sqrt{1-\cos^2 C}\;,$ bcz Largest side has…
juantheron
  • 53,015
5
votes
0 answers

Calculation of $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{m^2n}{3^n\left(m\cdot 3^n+n\cdot 3^m\right)}$

Calculation of $\displaystyle \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{m^2n}{3^n\left(m\cdot 3^n+n\cdot 3^m\right)}$. $\bf{My\; Try::}$ Let $\displaystyle S = \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{m^2n}{3^n\left(m\cdot 3^n+n\cdot…
juantheron
  • 53,015
5
votes
1 answer

prove equation equals $-1$

I was wondering if it was possible to prove that $\left( \frac{( a^2 - c^2 + (\frac{d}{c}a)^2-d^2)}{ \sqrt{(a-c)^2+(\frac{d}{c}a)-d)^2} \sqrt{(a+c)^2+(\frac{d}{c}a)+d)^2}} \right) = -1$ when $a,d,c \in [-1,1]$ and $|a|
Kate
  • 53
4
votes
1 answer

Roots of $2^{\frac x2} + (\sqrt{2} +1)^x = (5+2 \sqrt{2})^{\frac x2} $

I tried differentiating the equation to get minima and maxima, but failed to find the roots even there. Trial and error provides the answer=2, however, I'm searching for a proper method. Thanks in advance. :)
Harshal Gajjar
  • 1,664
4
votes
4 answers

how to find the solution to this system of equations

Given the system of equations: $ xy+xz=54+x^2 \\ yx+yz=64+y^2 \\ zx+zy=70+z^2 $ Need to find all of the solutions of $ x,y$ and $z$. Tried to sum up all three equations but got stuck with nothing to factorize.
gabi
  • 41
4
votes
3 answers

How can I find the point where two algebraic equations, in the form $y=mx+b$, intersect without graphing?

Suppose I have these two algebraic equations in the format $y=mx+b$: $$ y=2x+4 \\ y=3x+5 \\ $$ Now, by graphing these two algebraic equations on a coordinate plane, I find that they intersect at the point $(-1,2)$. Now, I find it annoying sometimes…
Ethan Bierlein
  • 1,242
4
votes
2 answers

If $f(x) = \sqrt{x}$, what is the domain of $f^4(x)$?

I am unclear if I should consider the function's domain before or after raising it to the power. My textbook gives the following definition of raising a function to a power: By $f^n$, we mean the function that assigns to $x$ the value $[f(x)]^n$.
Daniel Parker
  • 43
4
votes
3 answers

Proof of: if $x^2+y^2=2xy$ then $x=y$

I am trying to prove $x^2+y^2=2xy$ then $x=y$ What I have done is suppose $x^2+y^2=2xy$ then $x^2+y^2+(-2xy)=0\iff x^2+(-xy)+(-xy)+y^2=0 \iff (x+(-y))\cdot x+(x+(-y))\cdot-y=0 \iff (x+(-y))^2=0$ i then square root both sides but i'm not sure if…
Jhune
  • 111
4
votes
2 answers

How do I evaluate $\lim_{x \to -1} \frac {x^2+2x+1}{x^2+4}$?

I have determined so far that this is equal to $$\lim_{x \to -1} \frac {(x+1)(x+1)}{(x+\sqrt [4] {1})(x-\sqrt [4]{1})(x^2+\sqrt{1})}.$$ However, my numerator becomes $0$ if I substitute the limit. What am I doing wrong?
user137452
  • 1,043
4
votes
2 answers

Inequality proof of integers

My question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook. Page 36. Exercise 7. Let $n_1$ be the smallest positive integer $n$ for witch the inequality $(1+x)^n>1+nx+nx^2$ is true for all $x>0$. Compute…
George Apriashvili
  • 1,123
4
votes
3 answers

How to factor general equation of pair of straight lines with two variables and constant number at last?

How to factor the equation: $$x^2 + 2xy + y^2 - 2x - 2y - 15 = 0$$ I tried to solve this equation several times and looked for the reference about solving polynomial equations with two variables. I am confident to solve these type of equations…
user124672
  • 119
4
votes
3 answers

Use the law of logarithms to expand an expression

$$\log\sqrt [ 3 ]{ \frac { x+2 }{ x^{ 4 }(x^{ 2 }+4) } } $$ How is this answer incorrect? $$\frac { 1 }{ 3 } [\log(x+2)-(4\log x+\log(x^ 2+4))]$$
Cherry_Developer
  • 4,239
4
votes
1 answer

Algebra question I don't understand

It says: Solve the parameter $t$ when the equation $x^2=2t+4$ hasn't any real roots And well... I could solve for $t$, for $x$... but I don't get the actual point of the question. What should I do there? Thank you
Anna
  • 43
  • 2
4
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3 answers

Finding all solutions of an expression and expressing them in the form $a+bi$

$$6x^2+12x+7=0$$ Steps I took: $$\frac { -12\pm \sqrt { 12^{ 2 }-4\cdot6\cdot7 } }{ 12 } $$ $$\frac { -12\pm \sqrt { -24 } }{ 12 } $$ $$\frac { -12\pm i\sqrt { 24 } }{ 12 } $$ $$\frac { -12\pm 2i\sqrt { 6 } }{ 12 } $$ I don't know where to go…
Cherry_Developer
  • 4,239
4
votes
2 answers

proof regarding the commutativity of an arbitrary oddball binary operator?

My niece has shown me a problem for her advanced high school algebra class that I am personally finding fascinating, regarding the proof (or lack thereof) of the commutativity of a particular arbitrarily defined binary operator (basically, a…
Dan Nissenbaum
  • 141
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