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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 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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Limit of sum of the series

What would be the sum of following ? $$\lim_{n\to\infty} \left[\frac{1}{(n+1)^{2}} + \frac{1}{(n+2)^{2}} + \frac{1}{(n+3)^{2}} + \cdots + \frac{1}{(n+n)^{2}}\right]$$ I tried to turn it into integral : $\displaystyle\int…
Mojo Jojo
  • 463
4
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Methods for efficiently factoring the cubic polynomial $x^3 + 1$

$$x^3 + 1$$ factors as $$(x^2 - x + 1)(x + 1) .$$ It would have taken me a few minutes to identify this. What are the various approaches to determining rapidly that it is factorable, and factoring it?
Eric Johnson
  • 606
4
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3 answers

Find the value of $(a - b)^2$ given $a + b = 2$ and $a^2 + b^2 = 6$

$$a + b = 2\\ a^2 + b^2 = 6$$ Find the value of $(a-b)^2 $ My workings till I got stuck - $$(a-b)^2 = a^2 - 2ab + b^2 \\ = a^2 + b^2 - 2ab\\ = 6 - 2 ab $$ I'm stuck at how to find $ab$ . Can I get hints on how to find $ab$? Thanks a lot.
user307640
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If $x^2=y+z$, $y^2=x+z$ and $z^2=x+y$, prove

If $x^2=y+z$, $y^2=x+z$ and $z^2=x+y$, Prove that $$\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}=1$$. My…
Iaamuser user
  • 711
4
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2 answers

If $\ a_{n} = \frac{1000^n}{n!}$ for $n\in \mathbb{N}\;,$ Then $a_{n}$ is greatest when $n=$

If $\displaystyle a_{n} = \frac{1000^n}{n!}$ for $n\in \mathbb{N}\;,$ Then $a_{n}$ is greatest when $\bf{Options::}\;\;(a)\;\; n=997\;\;\;\; (b)\;\; n=998\;\;\;\; (c)\;\; n=999\;\;\;\;(a)\;\; n=1000\;\;\;\;$ $\bf{My\; Try::}$ Given $\displaystyle…
juantheron
  • 53,015
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5 answers

About Factorization

I have some issues understanding factorization. If I have the expression $x^{2}-x-7$ then (I was told like this) I can put this expression equal to zero and then find the solutions with the quadratic formula, so it gives me $x_{0,1}= 1 \pm…
Novato
  • 309
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Which of these statements about $Q=\frac{1}{100}+\frac{1}{101}+\cdots+\frac{1}{1000}$ is correct?

Pick the correct option regarding $Q$. $$Q=\frac{1}{100}+\frac{1}{101}+\cdots+\frac{1}{1000}$$ Pick one option: $Q>1\qquad$ 2. $Q\leq \frac{1}{3}\qquad$ 3. $\frac{1}{3}
Bazinga
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Solve $\sqrt[3]{7x+19}+\sqrt[3]{7x-19}=\sqrt[3]{2}$ by algebraic methods

I was trying to solve this equation without using calculus. Is it possible to be solved by elementary algebraic methods? $$\sqrt[3]{7x+19}+\sqrt[3]{7x-19}=\sqrt[3]{2}$$
user309912
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If $a+b=8$ and $ab+c+d = 23$ and $ad+bc=28$ and $cd=12\;,$ Then $abcd$

If $a+b=8$ and $ab+c+d = 23$ and $ad+bc=28$ and $cd=12\;,$ Then value of $(1)\;\; a+b+c+d=$ $(2)\;\; ab+bc+cd+da = $ $(3)\;\; abcd=$ My attempt: Let $x=a\;,b$ be the roots of $(x-a)(x-b)=0$ and $x=c\;,d$ be the roots of $(x-c)(x-d)=0.$ So,…
juantheron
  • 53,015
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3 answers

Two algebra questions

I have two questions that I need help with: 1) How many single digit even natural number solutions are there for the equation $A+B+C+D = 24$ such that $A+B > C+D$ A)20 B)11 C)16 D)24 2) Three positive real numbers $x,y,z$ are such that $x+y+Z = 1$.…
Dick Carlington
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1 answer

Find the equation of parabola passing through $(-1, 6), (1, 4), (2, 9)$

A(the?) equation of parabola is $y = ax^2 + bx + c$. That gives the equations below: \begin{align*} 6 & = a - b + c\\ 4 & = a + b + c\\ 9 & = 4a + 2b + c \end{align*} Then I simply solve for $(a, b, c)$ and substitute it into $y = ax^2 + bx + c$,…
user307277
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1 answer

finding the integral solutions

Prove that the equation $\sum\limits_{i=1}^n x_i^{-2} = 1$ has integral solutions for $n > 6$. I have no idea how to proceed, can someone give me a hint. Sorry for the mistakes in the text above.
thebeatles
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Doubt on displacement of a parabola(Again)

In another exercise is given: Find the parabola which is a displacement of $y = 2x^2 - 3x + 4$ which passes though the point $(2, -1)$ and has $x = 1$ as its symmetry axis. I've reduced the based equation to the form: $y = 2(x - \frac{3}{4})^2 +…
aajjbb
  • 1,065
4
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3 answers

Real values of $x$ satisfying the equation $x^9+\frac{9}{8}x^6+\frac{27}{64}x^3-x+\frac{219}{512} =0$

Real values of $x$ satisfying the equation $$x^9+\frac{9}{8}x^6+\frac{27}{64}x^3-x+\frac{219}{512} =0$$ We can write it as $$512x^9+576x^6+216x^3-512x+219=0$$ I did not understand how can i factorise it. Help me
juantheron
  • 53,015
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0 answers

If $f(n)= \binom{n}{0}a^{n-1}-..+(-1)^{n-1}\binom{n}{n-1}a^{0}$ ,Then $f(2007)+f(2008) $

If $\displaystyle a= \frac{1}{3^{223}}+1$ and $\displaystyle f(n)= \binom{n}{0}a^{n-1}-\binom{n}{1}a^{n-2}+...........+(-1)^{n-1}\binom{n}{n-1}a^{0}$ Then value of $f(2007)+f(2008) = $ $\bf{My\; Try::}$ Multiply both side by $a\;,$ We get $$af(n)…
juantheron
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