# Questions tagged [discriminant]

Discriminant of a polynomial $\;P\left(x\right) = a_{0} + a_{1}x + a_{2}x^{2} + \dots + a_{n}x^{n} \neq 0\,$ is defined as \begin{align} \Delta &= a_{n}^{2n-2}\prod_{ i < j } \big( r_i - r_j \big)^{2} = \left(-1\right)^{n\left(n-1\right)/2} a_{n}^{2n-2}\prod_{ i \neq j } \big( r_i - r_j \big) \end{align} where $\,r_1,\dots,r_n\,$ are roots of $P\left(x\right)$ (counting multiplicity)

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### discriminant to distinguish parallel line and double line degenerate conic sections

A real affine conic section is the zero locus in $\mathbb{R}^2$ of the quadratic form $$q(x,y)=ax^2+2bxy+cy^2+2dx+2ey+f=0.$$ We may understand this as the $Z=1$ affine patch of the locus in the ...
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### Finding the discriminant of a quaternion algebra

Consider the totally real number field $F=\mathbb{Q}(\zeta_{10}+\zeta_{10}^*)$. Consider the quaternion algebra $Q=(\frac{-1,-1}{F})$. How do I compute the discriminant of this algebra? I gave ...
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### convert a 1-dimensional set of points to a 2-dimensional parabola with explicit embedding

I am trying to rephrase to better understand concepts regarding discriminant functions for classification using explicit embedding. I report a very easy diagram found online that from 1-dimension ...
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### Smallest possible value of $k$ such that the roots of $x^2-127x+k=0$ are positive integers [closed]

In a triangle, two sides have equal lengths both shorter than the third side. The length of the three sides are all integers and all satisfy the equation $x^2-127x+k=0$, $k$ is a constant. Find the ...
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We have $\displaystyle{ a,b,c }$ real numbers such that $\displaystyle{ a^2+b^2+c^2>0 }$. Which condition must hold so that the equation $\displaystyle{ ax^2+bx+c=0 }$ has two different solutions? $... 1 vote 1 answer 74 views ### Finding Lattice points on a Cubic I want to study the rational points on a cubic. Eventually I found Nagell's algorithm from http://webs.ucm.es/BUCM/mat/doc8354.pdf, but I cannot immediately apply it because I don't know a rational ... 5 votes 1 answer 76 views ### Why do we need to include the factor$a_{n}^{2n-2}$in the discriminant of a polynomial? Question: Why do we need to include the factor$a_{n}^{2n-2}$in the discriminant of a polynomial? Here is the definition of the discriminant ($\Delta$) in terms of the roots$r_1,r_2,...$: $$\Delta=... 2 votes 1 answer 57 views ### Discriminant of f(x^n) I read about this post: Discriminant of f(x^n) for f a quadratic about \Delta f(x^n)=x^{2n}-bx^n+c if f(x) = x^2-bx+c is a quadratic. In particular, \Delta f(x^n) = n^{2n}c^{n-1}\Delta(f(x)).... 0 votes 2 answers 106 views ### Prove that the roots of g(x) are also real Suppose there are two equations$$f(x)=x^2+bx+c=0$$and$$g(x)=x^2+bx+c(x+a)(2x+b)=0$$It is given that f(x) has two real roots. Then prove that g(x) also has two real roots when a\in \mathbb{R}.... 1 vote 0 answers 51 views ### Find and classify bifurcation points for a cubic ODE I have an ODE which can be written as x' = g(x) = x^3 + px + q where p=-\frac{3c}{A} and q=-\frac{3d}{A} (we can assume c>0,d\neq. 0) and I am trying to find and classify the bifurcation ... 0 votes 2 answers 86 views ### Finding k such that the discriminant k^2 - 12k + 52 yields a quadratic with only one solution The given discriminant (not the quadractic) is k^2 - 12k +52 The exact question asks "find value of k when there is only one solution" - that would mean the equation would equal 0:$$k^... 0 votes 1 answer 94 views ### Nature of$\Delta$in polynomials$?$Suppose there is a function$f(x)=a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots +a_1x+a_0$and it is given that$a_n>0$and$f(x)>0\:\: \forall x\in \mathbb{R}$Then will it be accurate to say ... 1 vote 2 answers 91 views ### Is the theory of$\Delta\le0$, true in cubic functions$?$If$ax^2+\frac{b}{x}\ge c\forall x>0$where$a>0 \:\: , b>0$Show that$27ab^2\ge4c^3$My work: Let a function$f(x)$be $$ax^2+\frac{b}{x}\ge c$$ or we can rewrite$f(x)$as $$ax^3-cx+b\... 1 vote 1 answer 88 views ### Discriminant in a cubic extension There is a question: Compute the discriminant \triangle(1,\alpha,\alpha^2), relative to \mathbb{Q}(\alpha), where \alpha is a root of the reducible cubic x^3+px+q, p,q\in\mathbb{Q}. Is this ... 1 vote 0 answers 78 views ### Cauchy Schwarz Inequality for Vectors - Quadratic form Clarification In one of the proofs of Cauchys Inequality that reduces vector inner product to a quadratic form:$$0 \leq \langle u + \alpha, u + \alpha \rangle = \langle u , u \rangle + 2 \alpha\langle u ,v \... 3 votes 0 answers 460 views ### Why do we assume the discriminant to be greater than or equal to zero while calculating the range of a function? Since I don't have enough reputation to comment, I am asking this question again. I cannot understand why we can assume that the quadratic has real roots and then say$D\ge0$. The answer states that$...
So I have a math question regarding the least squares method. Given $$A=\begin{bmatrix}-1&2\\0&0\\1&-2\end{bmatrix}$$ $$b=\begin{bmatrix}1\\-2\\3\end{bmatrix}$$ I want to derive the least-...