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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How to factor $a^n - b^n$?

Wikipedia provides a proof, but I don't understand how: $$a^n - b^n = (a-b)(a^{n-1} + ba^{n-2} +\cdots + b^{n-1})$$ follows from $$x^{n-1} + x^{n-2} +\cdots + x + 1 = \frac{x^n - 1}{x-1}$$ Could someone explain to me how the summation of the the…
asdf
  • 335
18
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3 answers

Show that $(a+b+c)^3 = a^3 + b^3 + c^3+ (a+b+c)(ab+ac+bc)$

As stated in the title, I'm supposed to show that $(a+b+c)^3 = a^3 + b^3 + c^3 + (a+b+c)(ab+ac+bc)$. My reasoning: $$(a + b + c)^3 = [(a + b) + c]^3 = (a + b)^3 + 3(a + b)^2c + 3(a + b)c^2 + c^3$$ $$(a + b + c)^3 = (a^3 + 3a^2b + 3ab^2 + b^3) +…
Sawyier
  • 409
18
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13 answers

Proving the product of four consecutive integers, plus one, is a square

I need some help with a Proof: Let $m\in\mathbb{Z}$. Prove that if $m$ is the product of four consecutive integers, then $m+1$ is a perfect square. I tried a direct proof where I said: Assume $m$ is the product of four consecutive integers. If $m$…
Mister_J
  • 349
18
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6 answers

Formula for the square root of a number?

Is there a formula for the square root of a number, that only uses addition, subtraction, multiplication, or division?
user311559
18
votes
4 answers

Are the equations $2x - 2y = 11, x = y - 2$ unsolvable?

My 9th grade son had this math problem, which seemed unsolvable to me: $$2x - 2y = 11$$ $$x = y - 2$$ So we can use substitution to come up with: $$2(y - 2) - 2y = 11$$ Now distribute: $$2y - 4 - 2y = 11$$ Reduce $2y$ and $-2y$ cancel each other…
Scottie
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17
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3 answers

Solve this equation: $3^{3x} - 3^x = (3x)!$

I have this equation: $$3^{3x} - 3^x = (3x)!$$ We have to solve for $x$ integer. I did try to attempt but to no avail. I can't manipulate any side of this equation. I took common $3^x$ in the LHS of the equation and got a product: $(3^x) (3^{2x}-1)$…
user123
  • 197
17
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6 answers

Helping my daughter with her homework: solving an algebra word problem.

Three bags of apples and two bags of oranges weigh $32$ pounds. Four bags of apples and three bags of oranges weigh $44$ pounds. All bags of apples weigh the same. All bags of oranges weigh the same. What is the weight of two bags of apples and…
user55630
  • 187
17
votes
4 answers

Beginner questions about how functions work

Background First I apologize because the following are very elementary and annoying questions about functions. But I could sure use help. It's distressing to me that I'm trying to get better at math but I don't even understand a fundamental concept…
HJ_beginner
  • 1,715
17
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2 answers

square root / factor problem $(A/B)^{13} - (B/A)^{13}$

Let $A=\sqrt{13+\sqrt{1}}+\sqrt{13+\sqrt{2}}+\sqrt{13+\sqrt{3}}+\cdots+\sqrt{13+\sqrt{168}}$ and $B=\sqrt{13-\sqrt{1}}+\sqrt{13-\sqrt{2}}+\sqrt{13-\sqrt{3}}+\cdots+\sqrt{13-\sqrt{168}}$. Evaluate $(\frac{A}{B})^{13}-(\frac{B}{A})^{13}$. By…
Authawich Narissayaporn
  • 353
16
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3 answers

An incorrect method to sum the first $n$ squares which nevertheless works

Start with the identity $\sum_{i=1}^n i^3 = \left( \sum_{i = 1}^n i \right)^2 = \left(\frac{n(n+1)}{2}\right)^2$. Differentiate the left-most term with respect to $i$ to get $\frac{d}{di} \sum_{i=1}^n i^3 = 3 \sum_{i = 1}^n i^2$. Differentiate the…
Zach Conn
  • 5,043
16
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1 answer

Value of $\sin (2^\circ)\cdot \sin (4^\circ)\cdot \sin (6^\circ)\cdots \sin (90^\circ) $

How can I calculate the value of $\sin (1^\circ)\cdot \sin (3^\circ)\cdot \sin (5^\circ)\cdots \sin (89^\circ)$ $\sin (2^\circ)\cdot \sin (4^\circ)\cdot \sin (6^\circ)\cdots \sin (90^\circ)$ My solution: Let $$A = \sin (1^\circ)\cdot \sin…
juantheron
  • 53,015
16
votes
5 answers

Given that $x^y=y^x$, what could $x$ and $y$ be?

It's not too difficult to figure out that $x$ and $y$ can both be 1, and also $x$ can be 2 and $y$ can be 4 (and vice versa). But I can't rule out if there are other solutions. Does it have anything to do with inverse functions? Is there a way to…
yroc
  • 1,075
16
votes
1 answer

Solve $x^x=2x$ where $x\in\mathbb C$.

Solve $x^x=2x$ where $x\in\mathbb C$. Obviously, one solution is $x=2$. By WA, another solution is $x=0.346...$. How to solve it analytically, e.g. using Lambert W function? Thank you.
JSCB
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16
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If $a^2+a b+b^2=40$ and $a^2-\sqrt{a b}+b=5$, then find $a^2+\sqrt{a b}+b$

I was given this problem to solve with elementary methods (High School level). Knowing that $$\begin{align} a^2+a b+b^2 &=40 \\ a^2-\sqrt{a b}+b &=\phantom{0}5 \end{align}$$ find $$a^2+\sqrt{a b}+b$$ I tried to look for $\sqrt{a b}$, since the…
Raffaele
  • 26,371
16
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13 answers

$xy=1 \implies $minimum $x+y=$?

If $x,y$ are real positive numbers such that $xy=1$, how can I find the minimum for $x+y$?
user64528
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