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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.

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Could somebody please help me prove $\frac{a}1=a$ using the properties of real numbers introduced in elementary algebra?

I would like to prove the following useful property of real numbers: For every real number $a$, prove $\frac{a}{1} =a,$ where for all real numbers a and b: $\frac{a}{b}=a*\frac{1}{b}; b\neq0,$ and $\frac{1}{b}$ is the reciprocal or multiplicative…
user 85795
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Finding the roots of $f(x) = \left(\pi^x-\frac{1}{\pi}\right)\cdot\left(x^\pi\right)\cdot\left(\frac{1}{\pi^2}-\pi^x\right)$

One question from the IYMC 2023 was Find the roots of the function $$f(x) = \left(\pi^x-\frac{1}{\pi}\right)\cdot\left(x^\pi\right)\cdot\left(\frac{1}{\pi^2}-\pi^x\right)$$ Looking at it, I set each factor to $0$ and found $-1, -2$ and $0$ as…
Chris Daniel
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Find any two consecutive $100$-digit numbers whose sum of digits is a perfect square

Find any two consecutive $100$-digit numbers whose sum of digits is a perfect square. this is all I did and I am stuck lets say $abc \dots m$ is a $100$ digit number whose sum of its digits is a perfect square $$a+b+c+.....+m = k^2$$ lets say $abc…
Mia Genov
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Difficult algebra problem involving multiple square roots and proving a statement

Question is as follows: Prove that, when $$y=\frac{\sqrt{(1+x)}+\sqrt{1+2x}+\sqrt{x}}{\sqrt{(1+x)}-\sqrt{1+2x}+\sqrt{x}}$$ then $$4y^2(1+2x)^2=(y-1)^4x(x+1)$$ Despite several attempts at this, I have been unable to derive the required result. My…
GR L
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How should I revise my understanding of a "number" so that it makes sense for one number to represent three shapes in three different dimensions?

Take for instance the number 64. How is it that it can represent a line with a length of sixty-four units, a square with one side the length of eight units and a cube with one side the length of four units? I get that there are three different units…
ceyhun aytekin
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Confusion about meaning of this question. High school Algebra level.

The product of two numbers is 10. One of them is $a$. Express their sum in terms of $a$. The factors of 10 are 1,2,5,10. Thus $a \in {1,2,5,10}$ Thus the sum can be either 7 or 11. Now how does one express the sum in terms of $a$? How do I…
yiyi
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Three equations with a common positive root

If the equations $x^2+ax+12=0$, $x^2+bx+15=0$ and $x^2+(a+b)x+36=0$ have a common positive root, then $(b-2a)$ is equal to What I tried: Let $\alpha$ be common positive root of all equation. Then $$\alpha^2+a\alpha+12=0…
jacky
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Proof that 2 funtion of the type $f_t(x) = \frac{1}{t} \cdot e^{-tx²} $ don't intersect

so as the titles states I wan't to proof that no two functions $f_t$ of type $$f_t(x) = \frac{1}{t} \cdot e^{-tx²} \; \text{given that} \; t>0$$ share a point. This is a question from my textbook. However I come to the conclusion that there should…
Prankster
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If $f(x) = x^3 + ax^2 + bx + c$ has three distinct integral roots and $f(x^2+2x+2)$ has no real roots, then...

Suppose that $f(x) = x^3 + ax^2 + bx + c$ has three distinct integral roots and $f(x^2+2x+2)$ has no real roots. What is the minimum value of $a$? What is the minimum value of $b$? What is the minimum value of $c$? In the case when $a$, $b$ and…
dayDreams26
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Number of solutions of exponential equation

Can anyone tell me how to find number of solutions $(x+a)^x=b$? For example $(x+1)^x=-1$ has four complex solutions, $(x+3)^x=10$ has two solutions,one positive one negative, and $(x-4)^x=-10$ hasn't any solutions. PS.Sorry for my bad English,I hope…
Marcin Augustynowicz
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Aptitude Question

An Engine length $1000 $ m moving at $10$ m/s. A bird is flying from engine to end with $x$ kmph and coming back at $2x$. Take total time of bird traveling as $187.5$ s. Find $x$ and $2x$. My approach: $s+10$ and $2s-10$ are the speeds in both…
Vivek Sadh
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Find time when 2 cars meet?

There is a roadway between city A and B . A car P starts at 5:00 am from A and reaches B at 10:00 am. Another car Q starts from B at 7:00 am and reaches A at 9:00 am. Find the time when car P meets car Q ? I did as follows for car P , travelling…
Harish Kayarohanam
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Nice binomial identity

After inspiring by Cardano method for $3th$ degree equations, I noticed that it may be possible to express $(x+y)^n$ with only these terms: $x^n,y^n, xy,x+y $ Let's see some…
Ramin Rahimzada
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Need help solving this basic algebra question (addition and division)

Believe it or not I graduated with a BSc in Computing Science, but apparently that means nothing after being out of school for a year. The question is: $\frac{(c+n)}{(t+n)}=\frac{1}{4}$ Solve for $n$. My attempt: $c+n=\frac{1}{4}(t+n)$ $c+n=…
mpen
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Solve $\frac{x+25}{x-5}=\frac{2x+75}{2x-15}$

Solve $\dfrac{x+25}{x-5}=\dfrac{2x+75}{2x-15}$ $\Rightarrow \dfrac{(x-5)+30}{x-5}=\dfrac{(2x-15)+90}{2x-15} \ \ \ ...(1)$ $\Rightarrow 1+\dfrac{30}{x-5}=1+\dfrac{90}{2x-15}\ \ \ ...(2)$ $\Rightarrow \dfrac{30}{x-5}=\dfrac{90}{2x-15}\ \ \…
ronald christenkkson
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