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

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0
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4answers
54 views

Complete the square in the form $(px+q)^2+r, p > 0$

I'm going over some completing the square questions and I need to express, in the form: $(px+q)^2+r, p > 0$ the quadratic equation is $16x^2-8x+11$ I know how to get it in the form $p(x+q)^2+r$ ...
0
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0answers
31 views

Numerical method to find coefficient of quadratic function given target skewness

I have two samples, $X$ and $Y$, and for both I calculate the sample skewness Sk$(X)$ and Sk$(Y)$. My objective is to find $d$ such that, given $Z = X + dX^2$, Sk$(Z) = $ Sk$(Y)$. The coefficient of ...
0
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1answer
46 views

Solving x^4=a mod p, given a is a quadratic residue

Given prime number $p\equiv 1 \pmod 4$. Prove if $a∈F_p^×$ is a quadratic residue then the congruence $$x^4 ≡ a \pmod p$$ has either no solutions or four solutions. Give examples of each case.
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1answer
41 views

The quadratic $x^2-4kx+3k = 0$ has two distinct roots $m$ and $n$, where $m > n$ and $m - n = m^2+n^2$. What is the sum of all possible values of k?

I was trying to solve the following question: The quadratic $x^2-4kx+3k = 0$ has two distinct roots $m$ and $n$, where $m > n$ and $m - n = m^2+n^2$. What is the sum of all possible values of ...
0
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2answers
34 views

Let $k$ be a real number. Prove that if the equation $|x^{2} - 3x| = x-2+k$ has two distinct roots, then either $-1 < k < 2$ or $k > 3$?

The title is the problem. The condition "has two distinct roots" is ambiguous, but I assume it to be ``having exactly two distinct roots".
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2answers
37 views

Help finding the exact form through quadratic formula

Question: $$\frac{5m}{2}=2+\frac{1}{m}$$ I have attempted the question but my answer is not correct according to the book. $$\frac{5m^2}{2m}-2=0$$ $$5m^2-2=2m$$ $$5m^2-2m-2=0$$ When I placed my ...
3
votes
2answers
63 views

limits of $x$ for $( x-a)(x-b)>c$

After solving a quadratic inequality, I've ended up getting the solution in the format- $(x-a)(x-b)>c$, where $a,b,c$ are real constants, then how can I decide on the limits of $x$. will the ...
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0answers
41 views

How to show that $2^x$ is not in $O(x^2)$?

This is from Discrete Mathematics and its Applications I am working on 2e. I knew right off the bat from previous computer science courses that 2^x is not in O(x^2). I am having a difficult time ...
1
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5answers
151 views

Find the length of a leg of a right triangle, given the area and the length of the other leg

The length of one leg of a right triangle is $(x - 6)$ centimeters, and the area is $(\frac12 x^2 - 7x + 24)$ square centimeters. What is the length of the other leg? I think the equation that I need ...
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0answers
30 views

Why can't this inequality hold true for all n > k?

This is from Discrete Mathematics and its Applications I am having trouble with why the "inequality n <= C cannot hold for all n with n >k". Is this reasoning for this that there is no largest ...
3
votes
3answers
58 views

Is there a clever shortcut to showing that this function is in O(N^2)?

This problem is from Discrete Mathematics and its Applications I am currently working on 2a. I am trying to apply an example the book gave earlier Is there some similar clever trick I can apply ...
2
votes
2answers
55 views

Prove $ 17 $ is a square $ \pmod{2^k} $ for all $ k =1,2, \dots $

I'm trying to find a proof that $ 17 $ is a square residue $ \pmod{2^k} $ for all positive integers $ k $. I know some very general theorems such as the Quadratic Reprocity Law, but they work only for ...
2
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1answer
52 views

Geometry of the Quadratic Formula

I am well aware of proofs of the quadratic formula that show, by completing the square and other methods, that the quadratic formula is what it is. I have been scouring the Internet and other ...
2
votes
1answer
42 views

If I'm factoring $2p^2+p-10$ would the answer be $p(2p+5) -2(2p+5)$?

If I'm factoring $2p^2+p-10$ would the answer be $p(2p+5) -2(2p+5)$? And to check would I just distribute and see if it matches up to the original problem?
4
votes
2answers
36 views

proof $z \mid b$ and $w\mid b$

Question I'm working on: Let $a,b$ be integers with $b$ not equal to $0$. suppose $x^2+ax+b=0$ and $x=z,w$. If $z,w$ are integers, show that $z\mid b$ and $w\mid b$. Is it sufficient for me to show ...
2
votes
1answer
26 views

Solving a specific system of n non linear equations

I'm trying to solve a system of equations but I don't realy know how to tackle it. The equations all look as follows $a_1 x_1+b_1x_1x_2^2+c_1x_1x_n^2=d_1$ $a_2 x_2+b_2x_2x_3^2+c_2x_2x_1^2=d_2$ ...
0
votes
1answer
51 views

System of equations ac = ax + by and ac^2 = ax^2 + by^2

I have these two equations: $$ ac = ax + by$$ $$ac^2 = ax^2 + by^2$$ I have to figure out $\mathcal x$ and $\mathcal y$ using $\mathcal a, \mathcal b, \mathcal c$ which are variables but not set ...
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0answers
25 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 ...
1
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2answers
28 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 ...
17
votes
2answers
317 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 ...
1
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0answers
22 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 ...
1
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1answer
29 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 $$(\alpha-\beta)^2=(\alpha+\beta)^2 -4\alpha \beta$$ ...
0
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0answers
20 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 ...
4
votes
1answer
120 views

If $a^2+ab+b^2=c^2+cd+d^2$ then $a+b+c+d$ is not prime.

Let $a,b,c,d$ be positive integers such that $a^2+ab+b^2=c^2+cd+d^2$. Show that $a+b+c+d$ is not prime. My proof looks like this: $(a+b)^2 - ab=(c+d)^2-cd$ $(a+b)^2 - (c+d)^2=ab-cd$ ...
1
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2answers
42 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 ...
0
votes
2answers
55 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 ...
0
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0answers
42 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 ...
1
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2answers
77 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 ...
0
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4answers
222 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 ...
2
votes
2answers
40 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 ...
2
votes
4answers
49 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 ...
2
votes
1answer
45 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, ...
1
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1answer
74 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 ...
0
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0answers
61 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 ...
0
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0answers
23 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
73 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 ...
2
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3answers
101 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
435 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, ...
0
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0answers
27 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 ...
0
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1answer
38 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
87 views

Integer root of a quadratic [closed]

Determine the sum of all (distinct) positive integers $ n$ , such that for some integer $a$, $$ n^2 -an + 6a = 0. $$
1
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1answer
59 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 ...
0
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2answers
48 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)= ...
1
vote
1answer
52 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 ...
1
vote
1answer
125 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 ...
1
vote
2answers
254 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 ...
0
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0answers
20 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 ...
2
votes
2answers
69 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?
8
votes
5answers
243 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.
0
votes
2answers
37 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? ...