Mathematics Stack Exchange
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
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Solving for $x$ in an inequality

$$\frac{x^2-3x-2}{x^2+5x+6}<\frac{2-x}{x^2-4}$$ First, I got the things it couldn't be, which were: -3, -2, and 2 I thought to factor everything I could, and I got $$\frac{x^2-3x-2}{(x+3)(x+2)}<\frac{2-x}{(x+2)(x-2)}$$ I also cancelled out one value…
Alex
  • 337
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6 answers

$(1 + \sqrt{2})^{2020} = a + b \cdot \sqrt{2}$ what is the value of $a^2-2b^2$?

$a$ and $b$ are two integers such that: $$(1+\sqrt{2})^{2020} = a + b\sqrt{2}$$ What is the value of: $a^2 - 2b^2$? And I know that $a^2-2b^2=(a+b\sqrt 2)(a-b\sqrt 2)$ but the problem is Idk the value of $a-b\sqrt 2$ Note that newton's binomials or…
Pritchard
  • 107
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3 answers

If $4^x + 4^{-x} = 5$, what is $8^x + 8^{-x}$?

I found this competition math problem that I haven't been able to solve. If $4^x + 4^{-x} = 5$, find $8^x + 8^{-x}$. Setting $a = 4^x$, we see the problem is equivalent to saying: If $a + a^{-1} = 5$, find $a^{3/2} + a^{-3/2}$. So $a$ is the…
D_S
  • 33,891
5
votes
4 answers

How to tell if a cubic equation with positive coefficients has three real, negative roots

I have a cubic equation in $x$ $$x^3+bx^2+cx+d=0$$ where all the coefficients are positive. I know that with Descartes' Rule, the equation has no positive real roots, it either has 3 negative real roots or 1 negative real root and 2 complex…
mamotebang
  • 554
5
votes
4 answers

Understanding a step in: find the minimum value of $f(x)=\left|x-1\right| + \left|2x-1\right| + \left|3x-1\right| + \cdots + \left|119x - 1 \right|$.

Problem statement: What is the minimum value of $f(x)=\left|x-1\right| + \left|2x-1\right| + \left|3x-1\right| + \cdots + \left|119x - 1 \right|$? Quoted portion of a solution I am not understanding: "If we graph each term separately, we will notice…
Carm
  • 85
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3 answers

Stuck on Square Root Problem (yep, homework!)

Here's the simple question: Devon has a piece of poster board 45 cm by 20 cm. His teacher challenges him to cut the board into parts, then rearrange the parts to form a square. a) What is the side length of the square? b) What are the fewest…
m.smakg
  • 303
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Find $x$ in the equation $ax^3+bx^2+cx=d$

$\begin{equation} \tag{A} ax^3+bx^2+cx=d \end{equation}$ We can define Delta for quadratic equation to check whether the equation has answer or not....for $f(x)$ which contains powers higher of $2$ for Is there any method to see how many acceptable…
ehsan alizad
  • 87
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1 answer

Problem 2-7 in Spivak

One is asked to show that $ \sum\limits_{i=1}^{n} k^{p}$ (typo on $i$?) can always be written in the form $$\frac{n^{p+1}}{p+1}+An^{p}+Bn^{p-1}+Cn^{p-2}+\cdots.$$ The solution states: The proof is by complete induction on $p$. The statement is true…
user36546
5
votes
4 answers

Is there a simpler way to express the fraction $\frac{x}{x+y}$?

Can I simplify this expression, perhaps into two expressions $\frac{x}{x+y}$ or is that already simplified as much as possible?
politus
  • 153
5
votes
2 answers

Algebra where operators are assumed as variables.

Does there exists any form of Algebra where operators can be assumed as variables? For example: $$ 1+2\times3=7 $$ can be considered as: $$ 1\:(\mathrm{\,X})\:2\:(\mathrm{Y})\:3=7 $$ ?
user31230
5
votes
1 answer

Simplifying a complicated fraction (precalculus)

Simplify: $$\frac{\left(\dfrac{3x+x^3}{1+3x^2}\right)^2-1}{\dfrac{3x^2-1}{x^3-3x}+1}\; \div\; \frac{\dfrac{9}{x^2}-\dfrac{33-x^2}{3x^2+1}}{\dfrac{3}{x^2}-\dfrac{2(x^2+3)}{(x^3-x)^2}}$$ My…
ronald christenkkson
  • 1,623
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1 answer

Need proofread for deriving quadratic equation formula

How to Solve quadratic equation $$ax^{2}+bx+c=0$$ such as $$a \neq 0$$ Divide by a both side from the equation such as $$\frac{a}{a}x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$ $$\Rightarrow x^{2}+\frac{b}{a}x + \frac{c}{a} = 0$$ $$\Rightarrow x^{2} +…
bsdshell
  • 1,509
5
votes
3 answers

Solve the equation $\log_{1-2x}(6x^2-5x+1)-\log_{1-3x}(4x^2-4x+1)=2$

Solve the equation $$\log_{1-2x}(6x^2-5x+1)-\log_{1-3x}(4x^2-4x+1)=2$$ We have $$D_x:\begin{cases}1-2x>0\\6x^2-5x+1>0\\1-3x>0\\1-3x\ne1\\4x^2-4x+1>0\iff(2x-1)^2>0\iff x\ne\dfrac12\end{cases}\iff x\in(-\infty;0)\cup(0;\dfrac{1}{3})$$ Also the…
kormoran
  • 2,963
5
votes
6 answers

Calculate the integer part

I have to calculate the integer part of this: $$[(\sqrt{2}+\sqrt{5})^2]$$ I tried to write it like this: $$[2+5+2\sqrt{10}]=[7+2\sqrt{10}]=7+[2\sqrt{10}]$$ Any ideas?
marinaaaa
  • 183
5
votes
1 answer

Figure out domain and range

Is there a hard and fast way, step by step process to figure out domain and range? I don't know where to start and lack the insight to just know what it can and can't be. Thanks
user88720
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