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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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Solve the equation $8^x+3\cdot2^{2-x}=1+2^{3-3x}+3\cdot2^{x+1}$

Solve the equation $$8^x+3\cdot2^{2-x}=1+2^{3-3x}+3\cdot2^{x+1}$$ The given equation is equivalent to $$2^{3x}+\dfrac{12}{2^x}=1+\dfrac{8}{2^{3x}}+6\cdot2^x$$ If we put $a:=2^x>0$, the equation becomes $$a^3+\dfrac{12}{a}=1+\dfrac{8}{a^3}+6a$$ which…
kormoran
  • 2,963
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votes
1 answer

What operation can perform on multiple sets of any combination of elements so that each equals the mean of them all?

I have a set of integers $\{e_1, e_2, ..., e_n\}$ in which subsets of elements are taken, and operations are applied to each element so that the sum of elements in a set equals the mean of all subsets, $x$. For example, the sets $A = \{e_1, e_2,…
Ricil
  • 33
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Why does doubling the exponent very often double the amount of digits?

This is not an absolute rule as you easily find counter examples. However, in $[\![0;100]\!]$ there are 51 integers which follow this rule. Many times, the number of digits in $2^{n+1}$ is twice the number of digits in $2^{n}$.…
Biskweet
  • 51
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1 answer

Quadratic inequality with boundaries

Here is a very old high school exam question I am trying to solve (purely for interest only): If $a,b,c$ are real numbers such that $-1 \le ax^2+bx+c \le 1$ for $-1 \le x \le 1$ prove that $-4 \le 2ax+b \le4$ for $-1 \le x \le 1$ (Hint: Consider the…
TryinHard
  • 101
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2 answers

Question on radius problem

A car with 15 inch radius tires was driven on a trip of a distance equal to 400 miles. Two months later, with snow tires, the odometer indicated 390 miles for the same trip. Find the radius of the snow tires. This is how I attempted to solve it.…
1110101001
  • 4,188
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1 answer

Absolute value of an equation

If I have the equation $-5 = A + B$ and I decided to take the absolute value of both sides, would it evaluate to $\left|-5\right| = \left|A+B\right|$ or $\left|-5\right|=\left|A\right|+\left|B\right|$?
user1128422
  • 33
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Positive Solution of Exponential Equation

The equation is $2^{x+1}+2^{1/x^2}=6$. By inspection I see that $1$ is a solution. However, after trying to algebraically isolate for $x$, I was unable to deduce that $1$ is a solution. Given the simplicity of the value of the solution, I was…
allegro.sostenuto
  • 308
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2 answers

Whats wrong with this proof?

Theorem: $x$ is a real number with $x \neq 1.$ If $\frac {x^2+1}{x-1} =x$, then $x=-1$. If we suppose that $x=-1$. Then $\frac {x^2+1}{x-1} = \frac {(-1)^2+1}{-1-1} = \frac {2}{-2} = -1 = x$ I would say that the prove is not correct in the second…
user2051347
  • 313
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4 answers

Why can't we multiply both sides of $x-1=0$ by $x$ to create another solution $x=0$?

Consider the equation $x-1=0$. The obvious solution is $x=1$. However, what is stopping us from creating more solutions by multiplying both sides by arbitrary values? For example: $$ \begin{align} x-1 & = 0 \\ x\cdot(x-1) &= 0\cdot x \\ x\cdot(x-1)…
Monolith
  • 139
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votes
3 answers

Simplifying a Rational Expression

How do you simplify the following expression: $$\frac{x^3-27}{x^2+x-6}$$ Thanks for the help. Haven't seen this stuff since high school and I'm trying to help my younger sister out.
JustAnotherCoder
  • 133
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3 answers

Solve the equation $\sqrt{x^2+x+1}+\sqrt{x^2+\frac{3x}{4}}=\sqrt{4x^2+3x}$

Solve the equation $$\sqrt{x^2+x+1}+\sqrt{x^2+\dfrac{3x}{4}}=\sqrt{4x^2+3x}$$ The domain is $$x^2+\dfrac{3x}{4}\ge0,4x^2+3x\ge0$$ as $x^2+x+1>0$ for every $x$. Let's raise both sides to the power of 2:…
kormoran
  • 2,963
3
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0 answers

I'm confused: can u-substitution help convert from Cartesian to Polar?

Just for fun, I decided to try one method of converting a parabola to polar coordinates. Without loss of generality and assuming $a \neq 0,$ let's suppose our quadratic $y = ax^2 + bx + c$ can be factored as $$y(x) = a(x-r_1)(x-r_2)$$ Now I notice…
askquestions4
  • 483
3
votes
2 answers

Evaluate x given the equation $x + \sqrt x = \sqrt 3$

Given: $x + \sqrt x = \sqrt 3$ Evaluate: $x^3 - 1 / x^3$
user88226
  • 31
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votes
1 answer

Solving $a(b - c) < d$ for $a$ in two ways gives opposite results

I have $$a(b - c) < d$$ I want to solve for $a > ???$. I can divide off the parentheses to get $$a < \frac{d}{b - c}$$. But... let's say instead I subtract both sides $$- a(b - c) > - d$$ I can absorb the negative into the parentheses. $$a(c - b) >…
Tiny Tim
  • 423
3
votes
2 answers

Percentage related question

I was solving following question: Last year Elaine spent 20% of her annual earnings on rent. This year she earned 15% more than last year and she spent 30% of her annual earnings on rent. The amount she spent on rent this year is what percent of the…
noman
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