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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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Sum of real values of $x$ satisfying the equation $(x^2-5x+5)^{x^2+4x-60}=1$

I have this equation from this paper (Q.63) Find the sum of all real values of $x$ satisfying the equation-$(x^2-5x+5)^{x^2+4x-60}=1$. My attempt- $(x^2-5x+5)^{x^2+4x-60}=(x^2-5x+5)^{(x-6)(x+10)}$ So, $x=6$ and $x=-10$ makes the power $0$ and…
Soham
  • 9,990
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3 answers

Pre Calculus and Rate of Change

Sacha drains the water from a hot tub. The hot tub holds $1600$ L of water. It takes $2$ h for the water to drain completely. The volume of water in the hot tub is modelled by $V(t) = 1600 - \dfrac{t^2}{9}$, where $V$ is the volume in litres at $t$…
mehwish
  • 41
4
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Solve this logarithmic equation.

$2\log_6(x^{1/2}+x^{1/4}) = \log_4 x$ I don't have any good ideas how to start. Im stuck at this easy question, I don't know what is going on.
semeduk
  • 79
4
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4 answers

If $3^x +3^y +3^z=9^{13}$.Find value of $x+y+z$

Problem: If $3^x +3^y +3^z=9^{13}$.Find value of $x+y+z$. Solution: $3^x +3^y +3^z=9^{13}$ $3^x +3^y +3^z=3^{26}$ I am unable to continue from here. Any assistance is appreciated. Edited $9^{13} =3^{26}$ $=3^{25} (3)$ $=3^{25} (1+1+1)$ $=3^{25} +…
rst
  • 2,031
4
votes
3 answers

Simplifying the result formula for depressed Cubic

After understanding the Cardano's formula for solving the depressed cubic (of the form $x^3+mx=n$, of course), I tried to find the solution of the equation $$x^3+6x=20.$$ After plugging into the formula $$x=(n/2+\sqrt{ \frac{n^2}{4}+ \frac{m^3}{27}…
user304152
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1 answer

Any integer written as "polynomial" in irrational number

Does there exist an irrational number $x>2$ such that any positive integer $n$ can be written in the form $n=a_0+a_1x+a_2x^2+\dots$, where $a_i\in\{0,1,\dots,6\}$? Some irrational numbers like $\varphi=\frac{\sqrt{5}+1}{2}$ combine well to give…
user336268
  • 2,339
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2 answers

What does $\Big(\frac{(x+1)^2}{2}\Big)^n-\Big(\frac{(x-1)^2}{2}\Big)^n$ equal to?

Determine the highest degree term of the polynomial $$\Bigg(\frac{(x+1)^2}{2}\Bigg)^n-\Bigg(\frac{(x-1)^2}{2}\Bigg)^n, \quad n\in\mathbb{N}$$ The answer suggests that the highest degree term is equal to $\dfrac{2nx^{2n-1}}{2^n} +…
user302454
  • 653
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4 answers

How to simplify an expression like this: $(x^2+x^{-2}-2)^{1/2}$

Sorry, I am not sure how to do the maths mark-up on this site but hopefully the question will make sense. I should know how to do this, but I have got myself stuck! Can anyone help? $(x^2+x^{-2}-2)^{1/2}$
Magpie
  • 523
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1 answer

An operation on a whole is equal to an operation on each part?

I've been pondering this question as to how and when you can perform an operation on a complete "unit" and the answer is the same when performing the operation on the individual parts of the "unit" For example: $$ \sqrt{25\div4} =…
HDE K
  • 139
4
votes
3 answers

Doubling the Sides of the Cube

For the following question A cubical block of metal weighs 6 pounds.How much will another cube of the same metal weigh if its sides are twice as long ? Ans=48 Here is how I am solving it but I am getting 12 as the answer For the current cubical…
MistyD
  • 1,655
4
votes
2 answers

What can be the possible value of $a+b+c$ in the following case?

What can be the possible value of $a+b+c$ in the following case? $$a^{2}-bc=3$$ $$b^{2}-ca=4$$ $$c^{2}-ab=5$$ $0, 1, -1$ or $1/2$? After doing $II-I$, $III-I$ and $III-II$, I got, $$(a+b+c)(b-a)=1$$ $$(a+b+c)(c-a)=2$$ $$(a+b+c)(c-b)=1$$ I'm…
TheApe
  • 553
4
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1 answer

Prove that $|a|+|b|+|c|\le17$ if $|ax^2+bx+c|\le1$ for $0\le x\le1$

$|ax^2+bx+c|\le1$ for $0\le x\le1$ for real $a,b,c$. Prove the strict inequality $|a|+|b|+|c|\le17$. The best i could do was $|a|+|b|+|c|\le11+7\sqrt2$.
Random guy
  • 73
4
votes
1 answer

Roots equal constant

\[ \sqrt{a-b} + \sqrt{b-c} + \sqrt{c-d} + \sqrt{d-a} = K \] for some real constant $K$ and some real numbers $a$, $b$, $c$ and $d$. Find $K$. Again i apologize for the syntax and appreciate anyhelp thanks!
fosho
  • 1,491
4
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4 answers

How to solve $\cos(x)\cos(2x)\cos(4x)=1/8$

I have to solve $\cos(x)\cos(2x)\cos(4x)=1/8$. I can express it for $x$ only with $\cos(2x)=\cos^2(x)-\sin^2(x)$ and $\cos(4x)=\cos(2x+2x)$, but it only seems to become a really big expression and I have no clue how to proceed after... Any…
Rosalie Landry-Ouellet
  • 199
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If $p=3n$ and $n=6$, what's the value of $p$?

Is it $18$? I'm not to sure, that's why I have asked. I'm thinking if a number precedes a letter then you times it, is this correct?
greg
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