# Tag Info

### Why do we sometimes lose solutions when solving equations?

If that is the case why do we sometimes lose solutions when solving equations? We don't. If we take an equation and perform valid operations on it, we will never lose any solutions. If we look back ...

### Why do we sometimes lose solutions when solving equations?

One frequent cause of losing solutions is dividing by something that could be $0$. For example, you get an equation of the form $$a b = b c$$ and you jump to the conclusion $a = c$ without taking ...

### Why do we sometimes lose solutions when solving equations?

We lose solutions for the same reason that we lose keys, or lose our glasses: it is because we are not careful. If we are careful when solving equations, we will not lose solutions.

### Why do we sometimes lose solutions when solving equations?

TL;DR (short response) $$\text{operation"} \neq \text{function"}.$$ But if we remove references to functions from the question, the question is a good one. Discussion The problem with the ...

This is a good question. You are showing that $-n=(-1)n$, which is interesting in itself but not needed here. As you said, $-a$ is the number that added to $a$ gives you $0$. So the original equality $... 3 votes Accepted ### Why won't factoring lead to the same result as expanding when simplifying$(x+y)^2 - y^2 - x^2 + 2xy$? $$(x+y)^2 \color{green}{- y^2 - x^2} + 2xy=(x+y)^2 \color{red}{-(y+x)(y-x)} + 2xy \tag 1$$ It is a wrong claim, $$RHS{=(x+y)^2 - (y+x)(y-x) + 2xy\\ =(x+y)^2 - (y^2-x^2) + 2xy\\ =(x+y)^2 - y^2+x^2 + ... 3 votes ### How solve the problem f(x+2)=f(x)+4x+4 for any x Given that f(x+2)-f(x)=4x+4 here you can make a telescopic sum$$\require{cancel}\cancel{f(x+2)}- f(x)=4x+4\require{cancel}\cancel{f(x+4)}-\require{cancel}\cancel{f(x+2)}=4x+12\cancel{f(x+... 3 votes Accepted ### Help me find a mistake in showing$ \sqrt\frac{7-\cos 4x}{2} > -2\sin x $When dealing with an inequality of the form $$\sqrt[2n]{f(x)} > g(x),$$ you must take into account two possibilities. Either$g(x) < 0$, in which case it is sufficient to have$f(x)\geq 0$for ... 3 votes ### How to prove$(1+\sqrt{2})^n$is an irrational number for every$n \in \mathbb{N}$? Here's a very different approach that doesn't necessarily use induction. Show that$ ( 1 + \sqrt{2})^n + (1 - \sqrt{2})^n$is an integer$A$. If we absolutely wanted, we can show this via induction. ... 2 votes ### How do you write "The sum of all the elements in set A"? As long as we know that there is a way to add up elements of$A$, then writing$\sum_{a \in A} a$is perfectly acceptable, as pointed out in the comments—but if you want to have an actual routine (or ... 2 votes ### How solve the problem$f(x+2)=f(x)+4x+4$for any$x$Define$g(2x) = f(x)$, we find the difference sequence of$g$: $$0 \quad 12 \quad 32 \quad 60 \quad ... \\ 12 \quad 20 \quad 28 \quad ... \\ 8 \quad 8 \quad ... \\\ 0 \quad ...$$ Observe that the$4$... 2 votes ### How do we prove that$(x^2 + y^2)^2 + x^2 + y^2 < 1$is a disk? Let$D = x^2 + y^2$(so that$d = \sqrt{D}$is the length of the vector$(x, y)$). Then the constraint is that: $$D^2 + D < 1$$ But as$D$is non-negative,$D^2+D$is a monotone increasing ... 2 votes ### How do we prove that$(x^2 + y^2)^2 + x^2 + y^2 < 1$is a disk? If you let$x = r\cos(\theta)$and$y = r\sin(\theta)$where$r\geq 0$and$\theta\in[0,2\pi), then it results that: \begin{align*} (x^{2} + y^{2})^{2} + x^{2} + y^{2} < 1 & \... 2 votes Accepted ### Show\sum_{i=0}^n{i\frac{{n \choose i}i!n(2n-1-i)!}{(2n)!}}=\frac{n}{n+1}By noting that \begin{align*} i \frac{\binom{n}{i} i! n (2n-1-i)!}{(2n)!} \cdot \frac{n+1}{n} &= i \frac{(n+1)! (2n-1-i)!}{(n-i)!(2n)!} \\ &= \frac{\binom{i}{1}\binom{2n-1-i}{n-1}}{\binom{2n}{... 2 votes ### Show\sum_{i=0}^n{i\frac{{n \choose i}i!n(2n-1-i)!}{(2n)!}}=\frac{n}{n+1}$Supposing we start from $$\sum_{q=0}^n {n\choose q} \frac{q}{q+1} {2n\choose q+1}^{-1} = \frac{1}{n+1}.$$ The LHS is $$n \sum_{q=1}^n {n-1\choose q-1} \frac{1}{q+1} {2n\choose q+1}^{-1}.$$ Recall ... 2 votes ### How to determine argument from Euler’s form of complex numbers? For a same specific complex number, changing$r$will change the number. You can only maintain the number constant by changing its argument (by adding$2 \pi k, k \in \mathbb{Z}$) Changing$r$will ... 2 votes Accepted ### How to determine argument from Euler’s form of complex numbers? If$z = a + bi$then we have $$\begin{eqnarray} a + bi & = & r e^{i \varphi} \\ & = & r \left(\cos \varphi + i \sin \varphi \right) \\ \implies a & = & r \cos \varphi \\ \mbox{... 2 votes Accepted ### Function problem, pre-university level Guide: Since the quadratic function is even, the axis of symmetry is the y-axis, hence we know that 2a-3b=0. Also, we are told that the function is surjective, hence we know the minimum value is ... 2 votes ### Proving \frac{a+b+c}{a^2b+b^2c+c^2a+9}\ge \frac{abc+6}{abc+27} for a, b, c\ge 0; ab + bc + ca = 3 Here is a proof. If abc = 0, it is easy. In the following, assume that abc > 0. The desired inequality is written as$$\frac{(a + b + c)(abc + 27)}{abc + 6} - 9 - (a^2b + b^2c + c^2a) \ge 0$$... 2 votes ### prove using induction or in any other way that for all natural number n≥2, 3^n>3n+1 You want to prove that 3^{k+1}>3(k+1)+1, so it's not very good to write this inequality from the beginning. You want to end with it. Start from the left hand side only: 3^{k+1}=3\cdot 3^k>3(... 1 vote ### Finding the max of a function using f(x)=L(x)-E(x) where L(x) is linear and E(x) is exponential The question is ill-defined. Exponential functions have three degrees of freedom. Any exponential function can be written as$$E(x) = a(b)^x + c.$$Note that$$ab^{x+c} + d = ab^xb^c +d = \tilde{a}b^x ... 1 vote ### How do we prove that$(x^2 + y^2)^2 + x^2 + y^2 < 1$is a disk? Write it as$(x^2 + y^2 + 1/4)^2 < 5/4$. Since$x^2 + y^2 + 1/4 > 0$, this is equivalent to$x^2 + y^2 < \sqrt{5/4} - 1/4\$.
Since you tagged this as pre-calc, I will answer that level. A circle is a special case of an ellipse (a = b = r). The area formula as a result will be $$A = \pi \times r^2$$ The formula for the ...