# Tagged Questions

Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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### The cubic equation $x^3-5x^2+6x-3 = 0$ has solutions $\alpha$, $\beta$ and $\gamma$. [on hold]

The cubic equation $x^3-5x^2+6x-3 = 0$ has solutions $\alpha$, $\beta$ and $\gamma$. Find the value of $$\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}$$
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### Knowing that $a,b,c \in ℝ^*_+$ prove that $\frac{a+b}{a+b-c},\frac{b+c}{b+c-a},\frac{c+a}{c+a-b}$ don't belong simultaneously to the interval $(1,2)$

I have to solve the following problem but I don't know how to : Knowing that $a,b,c \in ℝ^*_+$ prove that $\frac{a+b}{a+b-c},\frac{b+c}{b+c-a},\frac{c+a}{c+a-b}$ don't belong simultaneously to the ...
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### How to solve the equation $xy = 1, x^{2x-y} = y^{2(x-y)}$

I have the following equation that I don't know how to solve: $$\begin{cases} xy = 1 \\ x^{2x-y} = y^{2(x-y)} \end{cases}$$ Here's what I've tried (but my mathematical instinct tells me that I didn'...
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### Sketching polynomial functions [on hold]

On a grid sketch two different polynomial functions which are of degree 5 and belong to the same family. write a general equation which describes the family of functions you have sketched.
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### If $(\sqrt{x^2-5x+6} + \sqrt{x^2-5x+4})^{x/2} + (\sqrt{x^2-5x+6} - \sqrt{x^2-5x+4})^{x/2}=2^{\frac{x+4}{4}}$, find $x$.

The main question is : If $(\sqrt{x^2-5x+6} + \sqrt{x^2-5x+4})^{x/2} + (\sqrt{x^2-5x+6} - \sqrt{x^2-5x+4})^{x/2}=2^{\frac{x+4}{4}}$, find $x$. My method : I first began by substituting $x^2-5x+5$ as ...
### How do you calculate what $k$ is? [on hold]
How do you calculate what $k$ is? $$\frac{d-1}{k} = \log \left(\frac{21}{100} d\right)$$ And how do I get the integer value?