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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Euler Four-Square Identity variants?

Is it well-known that there are an infinite number of Euler-type 4-square identities? Proof: $\begin{align} &{(x_1^2+ x_2^2+ x_3^2+ x_4^2) (y_1^2+ y_2^2+ y_3^2+ y_4^2)\,=\,z_1^2+ z_2^2+ z_3^2+ z_4^2}\\ &\text{where,}\\ &{z_1 \,=\, x_1y_1+a_1y_2+…
Tito Piezas III
  • 54,565
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factor the following expression $25x^2 +5xy -6y^2$

How to factor $$25x^2 +5xy -6y^2$$ I tried with $5x(5x+y)-6y^2$. I'm stuck here. I can't continue.
user155971
  • 1,515
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How is $x^2+1=(1/{x^2})[1-{1}/{x^2}+{1}/{x^4}-{1}/{x^6}+\cdots]$?

The author of my book writes: $$x^2+1=x^2\left(1+\frac{1}{x^2}\right)$$ $$=\frac{1}{x^2}\left[1-\frac{1}{x^2}+\frac{1}{x^4}-\frac{1}{x^6}+\cdots\right]$$ I do not understand the last step. How did the author write the last step. Please help.
M.S.E
  • 1,857
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2 answers

Square root of $x^3$

I understand the concept behind the expression $\sqrt{x^2} = |x|$. So, then why is the square root of $x^3$ NOT equal to $|x|\sqrt{x}$? Specifically, I can write $\sqrt{x^3}$ as $\sqrt{x^2\times x}$. Can I not now write this as $|x|\times…
user163862
  • 2,043
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Solve $x^4-3x^2+1=0$ in terms of cosine.

I put the equation in the form of a quadratic: $(x^2)^2-3x^2+1=0$ Then using the quadratic formula, $x^2=\frac{3\pm\sqrt{9-4}}{2}$ $x^2=\frac{3+\sqrt{5}}{2}$ and $\frac{3-\sqrt{5}}{2}$ $x=\pm\frac{1+\sqrt{5}}{2}$ and $\pm\frac{1-\sqrt{5}}{2}$ So…
yroc
  • 1,075
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2 answers

Calculating the percentage difference of two numbers

The basic problem is this: "I have this number x and I ask you to give me another number y. If the number you give me is some percentage c different than my number then I do not want it." Given that you will know x and c, how do you calculate…
Dan
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Please help me - algebra

It is considered positive integers $a, b, c$ with $a^2+b^2+a+b=abc$. Prove that $c\in\{3;4\}$.
user207637
  • 59
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Cyclic inequality for $n$ numbers

$a,b,c,x_1,x_2,x_3,...,x_n>0, a+b+c=1,\displaystyle \prod_{i=1}^n x_i=1 $ . Prove that $$(ax_1^2+bx_1+c)...(ax_n^2+bx_n+c)\geq1$$. I've tried just writing out as a product using the product sign and then to group certain parts of the product.…
Jackie Poehler
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2 answers

Solving a tricky equation

I'm out of practice with algebra, and I'm having a mind blank on how to solve for $t$ in the following equation. It's for some collision detection if you're wondering. $$\Bigl( \bigl(a-y-(qt)\bigr) \bigl(b-x-(rt)\bigr) \Bigr) - \Bigl(…
The_Denominater
  • 143
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2 answers

How is the plus-minus sign used for solving exponential and radical equations?

I noticed that for an equation such as $(4x+1)^2 = 289$, the plus-minus sign $\pm$ is used right after eliminating the squared expression $(4x+1)^2$ via square rooting. Yet, for an equation such as $\sqrt {4x-1}$ $=$ $\sqrt{x+2} - 3$, the…
logsofrhythms
  • 177
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Learning how to flip equations

I took Algebra and Geometry in high school, never thought I'd use them, then became a programmer. I guess I was wrong. To date, I have the hardest time taking equations and "flipping them," ie: rewriting an equation to find the reverse. For the…
Naftuli Kay
  • 231
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2 answers

Converting a polar equation to a rectangular one

$$r=\frac { 4 }{ 1+2\sin\theta } $$ Steps I took: $$(1+2\sin\theta )r=\frac { 4 }{ 1+2\sin\theta } (1+2\sin\theta )$$ $$r+2r\sin\theta =4$$ $$r+2y=4$$ $$(r+2y)^2=16$$ $$(r+2y)(r+2y)=r^2+4yr+4y^2$$ $$r^2+4yr+4y^2-16=0$$ My outcome doesn't seem to…
Cherry_Developer
  • 4,239
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2 answers

Beginner's algebra (real world application)

This may be too basic a question for this site, in which case I'm sure you'll all let me know. Here is my problem: I have a job worth $\$$40,000 composed of 6 units of x and 3 units of y. I'm trying to determine how much to charge for x and y, where…
Tom Auger
  • 175
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1 answer

If $g(x) = \max(y^2-xy)(0 \leq y\leq 1)\;,$ Then minimum value of $g(x)$

If $g(x) = \max\limits_{0 \leq y\leq 1}(y^2-xy)$, then minimum value of $g(x)$ $\bf{My\; try::}$ We can write $\displaystyle f(y) = y^2-xy = y^2-xy+\frac{x^2}{4}-\frac{x^2}{4} = \left(y-\frac{x}{2}\right)^2-\frac{x^2}{4}$ Now when $y=0\;,$ Then…
juantheron
  • 53,015
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2 answers

Three equations (almost linear), five unknowns, solve for three variables.

This problem doesn't seem to make sense to me. I have the following three equations: $$ S\alpha+1.06\beta + \mathcal{F} = S\\ T\alpha+1.06\beta + \mathcal{F} = T\\ 98\alpha+\beta + 0\mathcal{F} = 0 $$ where $S,T,\alpha,\beta, \mathcal{F}$ are…
nullUser
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