Questions on quadratic equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-h)^2+k$ or $y=a(bx+c)(cx+d)$.

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2answers
81 views

Query about a statement on the consequence of two quadratic equations having a common root

I have read an answer (in this site, but I lost the question number) saying something like the following:- If the quadratic equations F(x) = 0 and f(x) = 0 have a common root, then the quadratics are ...
3
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1answer
67 views

Solution to a System of Quadratic Equations

Problem: Solve for the values of a, b Equation 1: $$(x_1-a)^2+(y_1-b)=r^2$$ Equation 2: $$(x_2-a)^2+(y_2-b)^2=r^2$$ Where, $x_1, x_2, y_1, y_2$ and $r$ are all constant values For the ...
3
votes
1answer
178 views

$y =f(x) =(ax^2 + bx +c)/(dx^2+ex+f)$ We have to find the conditions for this it takes all real values.

$$ y=f(x)=\frac{ax^2+bx+c}{dx^2+ex+f} $$ We have to find the conditions for this it takes all real values. MY solution One approach is to equate it to y and for a quadratic of x and put discriminant ...
3
votes
1answer
113 views

Convexity of Quadratic equation Inequality?

Solving an inequality of the form $x^TAx\geq0$ or $x^TAx\leq0$ is straightforward. I mean we have to check if A is positive semidefinite or negative semidefinite. But what would be the solution to the ...
2
votes
1answer
51 views

Nonlinear first order ODE with quadratic in the derivative

This equation shouldn't be so hard, and yet I'm stymied. $$ \left( \frac{dw}{dz} \right )^2 + \alpha \frac{dw}{dz} + w \beta = 0 $$ with $w(0) = w_0>0$ $w(L) = 0$ for some known L and ...
2
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1answer
23 views

A question on multinomial theorem using binomial theorem

$(3x^2+2x+c)^{12}=\sum A_r x^r$ and $\frac{A_{19}}{A_5}=\frac 1 {2^7}$ Find $c$. I really have no idea what to do with this. This was on a test. I have studied only binomial theorem. So, please ...
2
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1answer
180 views

If I have an x intercept and a y intercept, how might I find the vertex of a parabola?

I have looked all over, and I have found different things here and there for figuring pretty much everything but that. What I have is an x chord and a y chord, but I need to find the vertex. This ...
2
votes
1answer
152 views

Finding a prime $p$ to solve a quadratic congruence $\pmod{p}$

I have a congruence of the form $$ax^2+bx \equiv -1 \pmod{p},$$ where $p$ is an odd prime and $a,b \in \mathbb{Z}$. Given $a$ and $b$, is there a general method to finding $p$ such that the above ...
1
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1answer
53 views

how to solve these two quadratic equations

Can someone help me find the solution for these two quadratic equations ? $ 2(z^2) \ - \ 3.023bz \ + \ 0.115(b^2) \ + \ 2.0814b \ + \ 0.142z \ - \ 0.5856 \ = \ 0 $ $ 6.0828(z^2) \ + \ 2.0414bz \ + \ ...
1
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1answer
36 views

Quadratic Equations - Mixed Roots

This is probably a silly question but why is it that when a quadratic equation has a single root it must be a repeated root. Why can't the second root be an imaginary root?
1
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1answer
84 views

what if geometric sequence + geometric sequence

I wrote a program that basicly can find the formula of the sequence that created with any-degree equation. For example if you give my program that sequence: [1926, 2811, 833240, 28778265, 398155842, ...
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1answer
54 views

Type of this Conic section

I want to determine, to which type the following Conic sections belong to: $$ \begin{align} \textrm{(i)}&\quad-8x^2+12xy-6x+8y^2-18y+8=0\\ \textrm{(ii)}&\quad5x^2-8xy+2x+5y^2+2y+1=0 ...
0
votes
1answer
71 views

Find pressure in a sinusoidal function

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 ...
0
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1answer
11 views

Quadratic expression

If $$\frac{a_0}{n+1}+\frac{a_1}{n}+\frac{a_2}{n-1}+\ldots+\frac{a_{n-1}}{2}+a_{n}=0,$$ then the maximum possible number of roots of the equation ...
0
votes
1answer
19 views

What is the minimum possible $ non $ integral value of a

Let a A subscript(m) (m=1,2,3,....p) be the possible integral values of a for which the graphs of $ f(x)=ax^2+2bx+b $ and $g(x)=5x^2-3bx-a$ meets at some point for all real values of b. 1) What is ...
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1answer
55 views

How to interpret coefficients in polynomial regression?

I am working on my thesis (study) about poverty incidence rate and its socio-economic factors using second-order polynomial regression without interaction. The final model in my study is ...
0
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1answer
50 views

Inequality challenge

I was studying inequations when I encountered this problem here. How can I find a region of values for m where this inequation is true? $$-3<\frac{x^2+mx-2}{x^2-x+1}>2$$ Thanks
0
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1answer
86 views

Solve a bit tricky system of equations

I want to solve the system for $x$, $y$ and $z$. Is there any smart trick to solve it? $$\begin{cases} 2a(ax+by)+2c(cx+dy)+2zx=0 \\ 2b(ax+by)+2d(cx+dy)+2zy=0 \\ x^2+y^2-1=0\end{cases}$$ $a,b,c,d \in ...
0
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1answer
99 views

When finding the dilation factor of $y = 3(2x - 3)^2 - \frac{1}{4}$, why must the brackets be expanded?

When finding the dilation factor of $y = 3(2x - 3)^2 - \frac{1}{4}$, why must the brackets be expanded? Why can't the outside factor of $3$ simply be used for the dilation factor from the ...
-1
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1answer
80 views

If $a,b,c \in R$ such that $c \neq0$ If $x_1$ is a root of $a^2x^2+bx+c=0, x_2$ is a root of $a^2x^2-bx-c=0 $ and $x_1 > x_2 >0$…

Problem : If $a,b,c \in R$ such that $c \neq0$ If $x_1$ is a root of $a^2x^2+bx+c=0, x_2$ is a root of $a^2x^2-bx-c=0 $ and $x_1 > x_2 >0$ then the equation $a^2x^2+2bx+2c=0$ has roots $x_3$ ...
3
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0answers
66 views

Fibonacci Quadratic Residue

After some research I have came up with a conjecture on fibonacci quadtratic residue: ...
3
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0answers
162 views

Second longest prime diagonal in the Ulam spiral?

Given the Ulam spiral with center $C = 41$ and the numbers in a clockwise direction, we have, $$\begin{array}{cccccc} \color{red}{61}&62&63&64&\to\\ ...
3
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0answers
79 views

Question about linearization

Given a data matrix $D\in\mathbb{R}^{N \times N}$ Can one construct another matrix $M$ that for all permutation matrices $Q^A$,$Q^B$, if $[\sum_i\sum_j (Q^A_{ij}D_{ij})]^2 \geq [\sum_i\sum_j ...
2
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0answers
89 views

Consider the quadratic equation $ax^2-bx+c=0, a,b,c \in N. $ If the given equation has two distinct real root…

Problem : Consider the quadratic equation $ax^2-bx+c=0, a,b,c \in N. $ If the given equation has two distinct real roots belonging to the interval $(1,2) $ then the minimum possible values of a is ...
2
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0answers
99 views

Proving an equality involving binomial coefficients and summations

Question: $$\sum_{k=0}^{n}\left ( -1 \right )^{k}\binom{2n}{k}\binom{2n-k}{2n-2k}=\sum_{2n}^{k=0}\binom{2n}{k}^{2}\left ( \frac{1+\sqrt{5}}{2} \right )^{2n-k}\left ( \frac{1-\sqrt{5}}{2} \right ...
1
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0answers
41 views

Quadratic Congruence in $\mathbb Z/2^n \mathbb Z$

Given the congruence $ax^2+bx+c \equiv 0 \pmod {2^n}$, how precisely does one go about finding its roots? I'm comfortable with quadratic congruence mod n with n odd, but 2's lack of a multiplicative ...
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0answers
51 views

Showing DO NOT exist GCD of $6$ and $2+2 \sqrt{-5}$ in $\Bbb Z[\sqrt{-5}]$.

Showing DO NOT exist gcd of $6$ and $2+2 \sqrt{-5}$ in $\Bbb Z[\sqrt{-5}]$. I tried it. Suppose $d$ is GCD of $6$ and $2+2 \sqrt(-5)$. then there exist $x,y \in \Bbb ...
1
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0answers
21 views

Variable quadratic functions

I have some doubts in proving the following:- C is the curve $$y = \frac 1 {k+1} [2x^2 + (k + 7)x + 4]$$, where k is a real number not equal to -1. Show that C always passes through two fixed points ...
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0answers
74 views

Quadratic equations with prime coefficients

I recently decided to go through old high school notebooks and I found something marginally interesting. I used to note down all kinds of things I came across, and I thought this might be useful for ...
1
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0answers
15 views

proving there exist another basis of non-degenerate quadratic space (V,B) other than the given basis

If {$v_i$} is a basis of non-degenerate quadratic space ($V,B$) (finite), prove that there exists another basis {$w_i$} such that $$B(v_i,w_j)=1 (i=j)$$ $$or 0(i \neq j)$$ Sorry for the ugly text ...
1
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0answers
58 views

How surfaces intersect in projective spaces

Consider this parametrization $$\phi:\mathbb{P}^1\longrightarrow\mathbb{P}^3$$ $$(t_0:t_1)\longmapsto (t_0^3: t_0^2t_1:t_0t_1^2:t_1^3)$$ Let $\mathcal{C}$ be the image of $\phi$. I've proved that ...
1
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0answers
65 views

Having trouble with finding a Quadratic Expression

I am having trouble trying to work out a quadratic expression for uni, being a external student its hard to find help. My problem is Perimeter = 1000m Part 1 Solve L in terms of w in regard to the ...
0
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0answers
24 views

A question using quadratic equations.

A ball is thrown down at 72km h-1 speed from the top of a building. The building is 125 metres tall. the distance travelled before it reached the ground is as follows... s = Uot + 1/5gt2 where Uo ...
0
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0answers
18 views

system of two quadratic equations with two variables

Is there a general way to solve exactly a system of this shape (the $a_i$ are constants): $$\begin{array}{cc}a_1x^2+a_2x+a_3y^2+a_4y+a_5=0\\ a_6xy+a_7x+a_8y+a_9=0 \end{array} $$ It comes from a ...
0
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0answers
10 views

Quadratic function as permutation of sequence

Say I have a $n \in \mathbb{N}$ and $$a_i := (1,2,...,2^n)$$ and two function $$f(i) = \sum_1^i i = \frac{i(i+1)}{2}$$ $$g(i) = f(i) mod 2^n$$ When I now look at a new sequence $$b_i = (a_{g(0)}, ...
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0answers
26 views

Convergence rate of $x_{k+1}=3x_k^2/n+3$

I've found the following claim in a slightly different form here (page 4, bottom of the left column) Starting from $x_0\le n/3$, the recurrence equation $$3\le ...
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0answers
14 views

How to minimize this quadratic function?

As described at page 3 of this document, I need to minimize the following quadratic function: $E(w,x,y,z) = \sum_i \frac{(w-T_i(x,y,z))^2}{1+|\Delta f(x_i,y_i,z_i)|^2} $ where $w=f(x,y,z)$ and ...
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0answers
14 views

Quadratics and function question

A quadratic function is given by ${h(x) = ax^2 + bx + c}$ where ${a}$, ${b}$, and ${c}$ are all nonzero real numbers. The function ${h(x)}$ intersects the x-axis at two distinct points and satsifies ...
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0answers
38 views

About a Variant of Ulam Spiral

Here I read about a variant on the Ulam spiral: [A] structure may be seen when composite numbers are also included in the Ulam spiral. [...] Using the size of the dot representing an integer ...
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0answers
25 views

Calculating $\arg\min_x (1-\Phi(x;\mu_1,\sigma_1^2)+\Phi(x;\mu_2,\sigma_2^2))$

I would like to find $x$ satisfying the following expression: $$\arg \min_x R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2)$$ where $$R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2) ...
0
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0answers
17 views

Ellipse equation. What does it need to be in order for $b > a$?

We have the quadratic equation: $$ax^2 + bx + cy^2 + dy + e$$ $a$ and $c$ are both negative or both positive. How can I, by looking at that only, determine whether $b$ (the length of the semi-minor ...
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0answers
40 views

Solving quartic equations

I am solving quartic equations by finding the intersections of two quadratics. The given quartic function is of the form $x^4 + px^2 + qx + r = 0$. I know one of the quadratic functions to be $y=x^2$. ...
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0answers
47 views

Given the matrix, find c such that det(D)=0 has a repeated solution

Given the matrix $D= \begin{vmatrix} 1 & x & x^2\\ 2 & c & 4\\ 3 & 2 & 1 \end{vmatrix} $ Find c such that $\det(D)=0$ has a repeated solution for $x\in R$. I got up to ...
0
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0answers
60 views

If $3x^{2}-2(a-d)x+(a^{2}+2(b^{2}+c^{2})+d^{2})=2(ab+bc+cd)$, then

If $3x^{2}-2(a-d)x+(a^{2}+2(b^{2}+c^{2})+d^{2})=2(ab+bc+cd)$, then $A.$ a,b,c,d are in G.P. $B.$ a,b,c,d are in H.P. $C.$ a,b,c,d are in A.P. $D.$ None of the above Tried writing the expression as a ...
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0answers
39 views

Complete the square

How would I complete the square for $y$ after completing the square for x below. Note that $y,x$ are vectors and not scalars. $$ (y-A^Tx)^TC(y-A^Tx)+(x-\mu)^TD(x-\mu)\\ ...
0
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0answers
56 views

If $a,b,c,k \in R, k >0$ and equation $px^2+qx+r=0$ has two real roots $\alpha , \beta $ such that $\alpha < -k$ and $\beta >k$…

Problem : If $a,b,c,k \in R, k >0$ and equation $px^2+qx+r=0$ has two real roots $\alpha , \beta $ such that $\alpha < -k$ and $\beta >k$, then $\forall \beta$ such that $ x \in R ,$ then ...
0
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0answers
286 views

If $ a+b+c=4 ; a^2=b^2+c=6 ; a^3+b^3+c^3=8 $ Then find the value of $a^4+b^4+c^4$

If $a+b+c=4$ $ a^2=b^2+c=6$ (this is not symmetric equation) $a^3+b^3+c^3=8$ Then find the value of $a^4+b^4+c^4$
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0answers
145 views

Sample Code to Generate Points on the Rim of a Randomly Rotated Cone : What's Going On Here?

Related to this question: http://math.stackexchange.com/questions/407897/randomly-generate-point-on-shell-from-3-points-2-angles-with-uniform-angle-dis I'm trying to reverse engineer the ...
0
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0answers
79 views

If $f(y)=ax^2+bx+c$, does this imply that $x=\frac{-b \pm \sqrt{b^2-4a[c-f(y)]}}{2a}$?

The equation $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$ is known as the quadratic formula and is the solution to the quadratic equation $ax^2+bx+c=0$. Sometimes I encounter equations such as $x=y^2-y$. Is ...
0
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0answers
48 views

In a quadratic extension, does the orthogonal complement of trace have a nice name?

The trace of an element in a quadratic extension is the sum of it with its conjugate. Is there a name for the difference between it and its conjugate?