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### How was the quadratic formula found and proven? [duplicate]

Possible Duplicate: Why can ALL quadratic equations be solved by the quadratic formula? History of Quadratic Formula How was the quadratic formula $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ found ...
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### Determine the value(s) of the ratio $x:y$ if $2x^2-xy-3y^2 = 0$ [duplicate]

I was asked to determine the value(s) of the ratio $x:y$ if $2x^2-xy-3y^2 = 0$. I didn't know what this meant, so I just solved the equation to get $x = -y$ or $x = \frac{3}{2}y$. What does the ...
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### How to derive the equation for x in a quadratic equation? [duplicate]

Possible Duplicate: Why can ALL quadratic equations be solved by the quadratic formula? How to derive this: $x = \frac{-b + {\sqrt{b^2 + 4ac}}}{2a}$ From this: $ax^2 + bx + c = 0$ I know ...
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I cannot understand how quadratic formula to solve for $x$ was derived. On this website, it explains the steps Following I understand but I cannot understand how they got $b/2a$ and why they ...
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### Where does the quadratic formula come from? [duplicate]

Everywhere I look, the $ax^2+bx+c$ portion of the quadratic formula is listed as given. Does anyone know where this comes from? Edit How can we prove that (x+y)^2 = ax^2+bx+c?
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$$0 = a(x^2 + \frac ba x) + c = a(x^2 + \frac ba x + \frac{b^2}{4a^2}) -\frac{b^2}{4a} + c$$ $$= a(x + \frac b{2a})^2 + c - \frac{b^2}{4a}$$ It is more than obvious that the above equation simplifies ...
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### Why can we prove mathematically that a formula to solve an (n+5) order polynomial does not exist?

I understand that the quadratic equation can solve any second order polynomial. Furthermore, equations exist for polynomials up to fourth order. However, without a graduate level degree and a deep ...
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### Why are polynomials defined to be “formal”?

Despite the fact that $\forall n, n^3 + 2n \equiv 0 \pmod 3$, I understand that $n^3 + 2n$ (considered as a polynomial with coefficients in $\mathbb Z/3\mathbb Z$) is not equal to the zero polynomial. ...
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### Simple explanation and examples of the Miller-Rabin Primality Test

Coming from an understanding of Fermat's primality test, I'm looking for a clear explanation of the Miller-Rabin primality test. Specifically: I understand that for some reason, having non-trivial ...
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### Sylvester's determinant identity

Sylvester's determinant identity states that if $A$ and $B$ are matrices of sizes $m\times n$ and $n\times m$, then $$det(I+AB)=det(I+BA),$$ where in the first case $I$ denotes the $m\times m$ ...
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### In generatingfunctionology, for a polynomial $P$ and a differential operator $D$, what does $P(xD)$ mean?

I'm working through some of the exercises in generatingfunctionology. One of the questions is to find the generating function where the $n$th term $a_n=P(n)$ for $P$ a polynomial. The answer is ...
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purplemath.com explains the quadratic formula. I don't understand the third row in the "Derive the Quadratic Formula by solving $ax^2 + bx + c = 0$." section. How does $\dfrac{b}{2a}$ become ...
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### How to find out X in a trinomial

How can I find out what X equals in this? $$x^2 - 2x - 3 = 117$$ How would I get started? I'm truly stuck.
### How to get from: $a^2 - a + 1 = 0$ to $a = \frac{1}{2}(1\pm\sqrt{1+4})$
Given that $a^2 - a + 1 = 0$, my book says: Therefor $a = \frac{1}{2}(1\pm\sqrt{1+4}).$ I have forgotten all the theory behind this.
### What are the benefits of using $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ to solve quadratic equations?
When I was in high school, they taught me to solve quadratic equations with this formula: $$x=\frac{\sqrt{4 \text{ac}+b^2}-b}{2 a}$$ EDIT: The original formula is this one: \$x = \dfrac{-b \pm ...