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

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15 views

Reference about quadratic forms with discriminant 1

When I am reading Serre's $A$ $Course$ $In$ $Arithmetic$, Chapter 5, it deals with $quadratic$ $forms$ of some vector space $V$, which can be viewed as an extension of an $abelian$ $group$ $E$ of ...
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2answers
18 views

How do I solve this equation?

I have an equation, where I need to find n, that I need help solving. I already cheated a little bit by using a CAS (Maple) to solve the equation, so i know what the result should be, but I need to ...
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0answers
9 views

Two modules with the same index have the same discriminant

In Serre's book $A$ $Course$ $in$ $Arithmetic$, it writes the following: Let $n=4k$, $k$ be a positive integer, $V=\mathbb Q^n$ be a $\mathbb Q$-vector space, endowed with the standard bilinear form ...
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2answers
154 views

Let $f(x)=x^2+12x+30$. Solve $f(f(f(f(f(x)))))=0$

Here is my solve, is it correct? I figured out that we can restate $f(f(x))$ as $((x+r)(x+s)+r)((x+r)(x+s)+s)$ thus $f(f(f(f(f(x)))))=0$ is $(x+r)^2(x+s)^2(4s+3r)(4r+3s)$ from vieta's ...
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0answers
14 views

Optimization of a Quadratic on a Linear Variety

We have a linear subspace $L_j := L[s_0,s_1,...,s_j]$ and a linear variety: $x_0 + L_j := [x_0 + y : y \in L_j]$ and a standard quadratic cost function $V(x) = a + b^Tx + 0.5x^TCx, \ \ \ C^T = C ...
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1answer
24 views

Finding an Expression for the Difference of Roots of the Quadratic Equation

Let the equation $ax^2+bx+c=0$ have the roots $\alpha$ and $\beta$, then what is $\alpha-\beta$ in terms of $a$, $b$, and $c$? Well, we may write $$(\alpha-\beta)^2=(\alpha+\beta)^2 -4\alpha \beta$$ ...
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0answers
11 views

Is there an efficent way to solve large systems of purely quadratic equations?

I have the following system of quadratic equations $$ b_1 = \sum_{k=1}^R x_{i_1, k} \ y_{j_1, k} $$ $$ \vdots $$ $$ b_p = \sum_{k=1}^R x_{i_p, k} \ y_{j_p, k} $$ where $i_1, \ldots, i_p \in ...
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0answers
46 views

$a^2+ab+b^2=c^2+cd+d^2$ prove that $a+b+c+d$ is a composite number for positive integers $a,b,c,d$ [on hold]

(Positive integers $a,b,c,d$ meet this condition $a^2+ab+b^2=c^2+cd+d^2$ )prove that $a+b+c+d$ As in the topic my proof looks like that; $(a+b)^2 - ab=(c+d)^2-cd$ $(a+b)^2 - (c+d)^2=ab-cd$ ...
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1answer
19 views

Quadratic equations with unknown [on hold]

My teacher gave me a Quadratic equation, which has unknown in it 2x^2 + 3x - k=0 I only know that D > 0 And i need to find K How can i do that?
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2answers
40 views

polynomial of $4^\text{th}$ degree, prove

There is a polynomial $f$ of integer coefficients such that $\text{deg(f)} \geq 4$. Let's assume that there are four integers $a,b,c,d$ for which $f(a)=f(b)=f(c)=f(d)=5$. Prove that there is no ...
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2answers
24 views

Finding the value of a constant given an equation where the sum of the roots is -3

I am to find the value of h given the equation 3hx^2 - 2x +5xh = 3. The sum of the roots of the polynomial is -3. I am having ...
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0answers
33 views

quadratic formula for polynomials with variable coefficients

I have trouble calculating equations like the one in last comment in the first answer; Solve system of 3 equations there are variable coefficients which I can calculate using quadratic formula - if ...
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2answers
68 views

Is it true that $a$ can't be zero in the quadratic function $y=ax^2+bx+c$?

I read that for $y=ax^2+bx+c$ is a quadratic function where $a\neq0$, but is it true that $a$ really can't be zero? I think it is because if $a$ was zero, there wouldn't be a parabola. There would ...
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0answers
26 views

Inexistence of Formulas for general solution of nth degree equations [closed]

For n>=5, there exists no direct formula to obtain roots of the equation (say in x) ,relating the coefficients (like the well known quadratic, cubic equations). Can you provide a short proof as to why ...
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4answers
43 views

How can I find the vertex of a parabola using only $x$ intercepts.

My teacher gave me this problem where I did a long jump and recorded the distance I went. He then asked us the height. My distance was 80 inches so the x-intercepts are $0,0$ and $80,0$. My question ...
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0answers
40 views

If I know what my $x$-intercepts are how can I find my vertex? [closed]

If I jump 80 inches how can I figure out how high I jumped? And is there an equation I can use to figure out the vertex of the parabola of my jump? The main question is what is the vertex of my jump ...
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0answers
52 views

If I jump 80 inches how high did I jump? [closed]

If I know I jumped 80 inches and those are the two points I have how can I figure out how high I jumped? And how can I use a formula?
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2answers
36 views

Prove relations between the roots of 3 quadratic equations

Let $x_1, x_2$ be the roots of the equation $x^2 + ax + bc = 0$, and $x_2, x_3$ the roots of the equation $x^2 + bx + ac = 0$ with $ac \neq bc$. Show that $x_1, x_3$ are the roots of the ...
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4answers
39 views

Use substitution to solve for $x$ in $\frac{1}{2-\sin x}=\sin x$

Use substitution to solve for $x$ in the following equation: $$\frac{1}{2-\sin x}=\sin x$$ This is what I have done so far: $$\sin^2x-2\sin x+1=0$$ $$\arcsin(1)=\frac{\pi}{2}=x$$ The correct ...
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1answer
39 views

Using $x = y - b/2$ to solve a quadratic equation

I've been reading a book called Mathematics for the Nonmathematician, and it presents a solution to quadratic equations of the form: $x^2 + bx + c = 0$ which relies on coming up with a new formula, ...
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1answer
27 views

Graphing Quadratic Function Describing a Parabolic Arch [closed]

An architect decides to use a parabolic arch for the main entrance of a science museum. In one of his plans, the top edge of the arch is described by the graph of $y=-\frac{1}{4}x^2+\frac{5}{2}x+15$. ...
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1answer
63 views

Find sum of the roots of quadratic polynomials [closed]

The zeroes of a quadratic polynomial $x^2+ax+b$ are $c$ and $d$ and the zeroes of a quadratic polynomial $x^2+cx+d$ are $a$ and $b$. Find the value of $a+b+c+d$. The thing doesn't make sense how ...
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0answers
5 views

Convert Bearing Degrees to Slope

I'm working on a mapping application that sketches a graphic on the map. The user will input the distance of a line segment and provide a bearing for the line segment. For example, a line segment is ...
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0answers
19 views

How is Lagrange's $2\sqrt{D}$ bound on partial denominators proven for periodic regular continued fractions of quadratic irrationals

For the quadratic surd: $$ \zeta = \dfrac{P + \sqrt D}Q $$ the wikipedia article on periodic continued fractions mentions that Lagrange proves that the largest partial denominator of a regular ...
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1answer
47 views

Computing Coefficients of a Quadratic Equations given Definite Integral, Height and Rate of Change

This question follows on from my previous question I asked at Computing Coefficients of a Quadratic Equation given definite Integral This time however I have two free variables. For the quadratic ...
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3answers
95 views

$\sin^n a + \cos^n a = 1$ is only true when $n=2$

Prove that $$\forall a\in\mathbb R:\quad\sin^n a + \cos^n a = 1$$ is only true when $n=2$
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1answer
42 views

How can I find the equation of a parabola only given it's x-intercepts?

I received a problem in my math class the other day that left me stumped. The problem went something like this. Mr. Lots-O-Cash would like to order a parabola that passes through the points $(-4, ...
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0answers
24 views

Find two integers $a, b$ for given integer $c$, so that $c=a^2\pm b^2$

Given a positive integer $c$: Find two other positive integers $a$ and $b$, so that $c=a^2 + b^2$ and/or $c=a^2 - b^2$. I've already got a solution for any odd $c$: $c = (x+1)^2 - x^2 = 2x + 1$ so ...
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1answer
30 views

Quadratic form on Vector Bundle

A quadratic form of a vector space $V$ over a field $\mathbb{F}$ is a bilinear symmetric map $V\times V \rightarrow F$. How does one define a quadratic form over a vector bundle.
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2answers
77 views

Integer root of a quadratic [closed]

Determine the sum of all (distinct) positive integers $ n$ , such that for some integer $a$, $$ n^2 -an + 6a = 0. $$
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1answer
35 views

How do you calculate the coordinates a quadratic curve follows?

I'm a programmer, and terrible at maths. Usually, I try Google or my math-addict co-worker for problems like this, but Google searches show nothing and my co-worker is on vacation for a few weeks. I ...
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2answers
31 views

Computing Coefficients of a Quadratic Equation given definite Integral

For the quadratic function $$-ax^2 + 1$$ an upside down parabola with $y(0) = 1,$ is there a way to compute a such that the definite integral of $y$ between the roots ($x_1, x_2: f(x_1) \land f(x_2)= ...
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1answer
51 views

Evaluating$\int\frac{1}{(x^2-1)^2}$

This is the integral: $\int\frac{1}{(x^2-1)^2}$ I have tried several ways to solve this but I always end up that last parameter equals 1 and all others equals 0 so I end up where I started. Examples ...
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1answer
24 views

Locus of the centers of the circles tangent to a given line and circle

Say you are given a circle $C$ and a straight line $l$ exterior to the circle. How to describe the set of centers of circle that are tangent to both the $C$ and $l$? I have no idea how to proceed. My ...
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2answers
52 views

Optimization: maximum area of a triangle under a parabola

Optimization: maximum area of a triangle in a parabola Inside a curve ($x^2-25$ - Parabola) a triangle is drawn with A as the vertex at the origin and the line joining points B and C lie on the ...
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0answers
18 views

Finding a homeomorphism between quadratic polynomials

I would like to represent a quadratic polynomial $f(x)=ax^2+bx+c$ as $$f=\phi\circ f_\lambda \circ \phi^{-1},$$ where $f_\lambda(x)=\lambda x(1-x)$ with $\lambda = 1+\sqrt{(b-1)^2-4ac}$. Is this ...
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2answers
36 views

Does the quadratic formula always work for a quadratic?

Does the quadratic formula always work for a quadratic? If it does, why are the factors sometimes imaginary numbers?
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5answers
209 views

Solving $2^{2x+1} - 2^{x+4} = 2^3 - 2^x$

$$2^{2x+1} - 2^{x+4} = 2^3 - 2^x$$ How can I solve an exponential equation that has many terms as the one above. Include more than one method if available.
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2answers
31 views

Find constant a where quadratic equation equals zero

I need to find the $a$ where $$(a-2)x^2 + (a^2 - a - 2)x + 2a^2 -4a = 0.$$ Ok, it is easy to tell that $a$ must equal 2 but... how can I find it if it's not so obvious? Do I have to take discriminant? ...
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1answer
34 views

reconstructing a quadratic equation from roots

I have this quadratic equation $$2x^2+x-3=0$$ that I wish to reconstruct from its roots. $$D=b^2-4ac=25$$ $$x_1=\frac {-b\pm \sqrt D} {2a} = 1 \text{ and } \frac {-2} 3$$ Now, I've always learned that ...
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1answer
44 views

find a quadratic polynomial p ( x ) and a number n such that p ( x ) and a number $n \pmod n $ has at least 2015 roots?

I understand what the question is asking for, but I don't know how to prove my answer. Let's say I took an equation of the form: $x^2+ 6x+ 8 \equiv0 \pmod {15}$. The first four roots are ...
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1answer
43 views

To find positive roots of a quadratic equation

Find all values of $a$ so that: $$x^2- ax+(4a+1)= 0$$ has both roots positive. I have been working hard and long on known facts but am unable to crack this one. Finally saw this site and hope for ...
2
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3answers
102 views

Better way of factorising $x^2-a^2+x+a$

I am currently at the subject factorisation and I have the following problem: Fully factorize: $$ {x^2}-{a^2}+x+a $$ What I did was the following: Create a common factor: $$ x({1^2}+1)-a(1^2-1) $$ ...
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2answers
43 views

Conditions for roots of quadratic equation

If I have a quadratic function $f(x)=ax^2+bx+c$, what are the conditions that should the numbers a,b and c satisfy so that the equation $f(x)=0$ has real roots $x_1$ and $x_2$ such that ...
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2answers
41 views

Nature of The Roots of The Quadratic Equation $(a-1)x^2+(4a-2)x+4a+1=0$ [closed]

For which values of the real parameter $a$ are the roots of the quadratic equation: $$ (a-1)x^2+(4a-2)x+4a+1=0 $$ a) Real b) Positive
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2answers
49 views

What is the equation whose roots are 2+√3 and 2-√3? [closed]

Please solve these equation. How to solve these I don't know.
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1answer
165 views

Solve $x^4+3x^3+6x+4=0$… easier way?

So I was playing around with solving polynomials last night and realized that I had no idea how to solve a polynomial with no rational roots, such as $$x^4+3x^3+6x+4=0$$ Using the rational roots test, ...
2
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1answer
30 views

Condition that a quadratic function may be resolved into two linear factors

If we are given a general quadratic function in $x, y$, there is a condition that it can be resolved into two linear factors of the form $ax+by+c$. I found the following proof for this. If we have a ...
0
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1answer
24 views

Find roots for an equation with quadratic, linear and log terms?

I'm wondering if there exists a closed-form or analytic expression for the roots of an equation of the form $ax^2 + bx + c\log x=0.$ considering the natural $\log$. Wolfram alpha is leading me to ...
1
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1answer
22 views

Determine which quadratic congruences have solutions

I need to determine which congruences of the form $ax^2+bx+c\equiv0\pmod{2}$ have solutions. What I know is that $a,b,c$ are all odd. I admit I have no clue how to begin on this one. This is at the ...