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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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Indeterminate equations: Special cases

[I had trouble deciding how to tag this question, so please edit if there's a better classification.] Some special types of indeterminate equations are solvable. for example, consider this example from the…
ankush981
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Are there any positive integers $n > 3$ such that $a_1^n+a_2^n+\cdots+a_n^n-na_1a_2\cdots a_n$ is factorable in integers? If not, then why?

I know that if $n = 1$ then $a_1^n+a_2^n+\cdots+a_n^n-na_1a_2\cdots a_n$ becomes $a_1-a_1 = 0.$ If $n = 2$, then the expression becomes $a_1^2+a_2^2-2a_1a_2$, which is $(a_1-a_2)^2.$ If $n = 3$, then the expression becomes…
py_math
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Finding $\alpha$ and $\beta$ such that the graph of $g(x)=2\sqrt{x-\alpha}+\beta$ passes through $(3, -2)$, $(4,0)$, $(7,2)$

I've a homework problem that I'm struggling to solve. I need to find the value of $\alpha$ and $\beta$ that will make the function $$g(x) = 2\sqrt{x-\alpha} + \beta$$ pass through $(3, -2), (4,0), (7,2)$. The best I got was try to convert the…
Hanan
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If working together, A and B complete 2 projects in 12 hours, how do you show that working separately, it may take over 24 hours to do 2 projects?

Here is a semi-algebraic explanation. Suppose $A$ works faster than $B$. Imagine that $A$ and $B$ start working on their separate projects at the same time. $A$ completes his project in time $t < 12$ hours. Suppose $A$ now helps $B$ to complete…
user1153980
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If $x^2 + y^2 = 1$, then find the value of $(3x-4x^3)^2+(3y-4y^3)^2$ where $x,y\in \mathbb {R}$ .

If $x^2 + y^2 = 1$, then find the value of $(3x-4x^3)^2+(3y-4y^3)^2$ where $x$, $y \in \mathbb {R}$ . What I've tried : I thought $x$, $y$ as $\sin \theta$ and $\cos \theta$ . But, it doesn't seems to work and I'm unable to see any particular…
Ash_Blanc
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Solving $(x+1)(4x-3) = 2(x+1)(2x+3)$. Why does dividing by $x+1$ cause problems?

I was just now trying to solve the following simple equation: $$(x+1)(4x-3) = 2(x+1)(2x+3)$$ In my naivety I thought dividing by $(x+1)$ would shorten the equation and lead to a quick solution, but unfortunately it lead to x disappearing from the…
MarcVolkert
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If $x+y=m$ and $x-y=n$, show that $16(x^4-7x^2y^2+y^4)=(5m^2-n^2)(5n^2-m^2)$

If $x+y=m$ and $x-y=n $, Show that $16(x^4-7x^2y^2+y^4)=(5m^2-n^2)(5n^2-m^2)$ $x^4-7x^2y^2+y^4=(x^2-y^2)^2-5x^2y^2$ $\Rightarrow [(x+y)(x-y)]^2-\dfrac{1}{5}(5xy)^2$ Then I'm stuck. The problem is the $5xy$ term has an odd coefficient, so I think no…
ronald christenkkson
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Compute $ax^3+by^3$ given $ax^2+by^2$ and $ax+by$.

Given are positive real numbers $A$ and $B$ and positive integers $a$ and $b$ such that $$ \begin{aligned} ax+by &= A\\ ax^2+by^2 &= B.\end{aligned} \tag{*}$$ What are the possible values of $$ ax^3 + by^3 = C?$$ Think of it as $a+b$ numbers, $a$…
Myself
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What is the distinction between variable and number in these two expressions?

The author here discriminates between the two functions as a is a fixed number, and x is a variable. I am confused as to this distinction, as it seems the reason why we would need to use 'a' instead of a number is to account for many different…
Princess Mia
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Find the removed term from an arithmetic series, given the sum of the other terms

We are given a sequence made up of pairs of numbers, $(1, 2), (3, 4), …,(x, x+1)$, out of which we must remove one pair. The sum of the numbers which have not been removed is 156. Calculate the sum of the removed numbers. This is what I got so…
Zero
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calculate ratios of two rice

Joseph bought two varieties of rice, costing $5$ cents per ounce and $6$ cents per ounce each, and mixed them in some ratio. Then he sold the mixture at $7$ cents per ounce, making a profit of 20 percent. What was the ratio of the mixture? my…
dato datuashvili
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calculate ratios of elements

let us consider following problem : Two alloys A and B are composed of two basic elements. The ratios of the compositions of the two basic elements in the two alloys are $5 : 3$ and $1 : 2$, respectively. A new alloy X is formed by mixing the two…
dato datuashvili
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calculate gradient

I'm trying to understand differentiation. At 9:45 the kind sir is filling in numbers in this video. My question is, where is he getting the gradient numbers from when hes filling in the table? I know it can be calculated by using nx^n-1, but I don't…
user1534664
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When is it valid to convert "$\le$" or "$\ge$" to "$=$"?

I don't understand why the conversion of ≤ to = in this proof in Spivak's Calculus is legitimate. (The conversion in the previous line from = to ≤ makes sense to me.) Could someone please explain or elaborate on how $(2)$ follows from $(1)$…
Lin
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How to simplify a term with $x$ and $y$ multiplied together

this may seem very dumb, but for some reason I can't quite remember how to simplify an equation of the following form in order to get $x$ in terms of $y$ or vice-versa $$ x^2 -2xy = y - y^2 $$ i keep getting stuck at the $2xy$ term
JoeVictor
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