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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32 views

Counting the solutions of a quadratic equation

I have read that a non-singular conic will contain $p+1$ points on the finite field $\mathbb{F}_{p}$, but there is a exercise on Silverman's Rational Points on Elliptic Curves, p.142, 4.8 that tells ...
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2answers
29 views

Bound on Coefficients

For real $a,b,c$ the following holds $|ax^2+bx+c|\le 1 ; \forall x\in [0,1]$.Show that $|a|+|b|+|c|\le 17$. Cant show that the equality holds.I always get the lesser bounds.
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1answer
56 views

A sequence of quadratic polynomials

Q. Let $p_n(x)=a_n x^2+b_n x$ be a sequence of quadratic polynomials where $a_n,b_n \in \Bbb R$ $\forall n \ge 1$. Let $\lambda_0$,$\lambda_1$ be distinct nonzero real numbers such that $\lim_{n ...
3
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2answers
132 views

System of quadratic equations over field of size 2

I am working on system of quadratic equations. \begin{cases} (\alpha_1^1x_1+\ldots+\alpha_n^1x_n)(\beta_1^1y_1+\ldots+\beta_m^1y_m)=0\\ \ldots \\ ...
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2answers
26 views

Graphing Parabolas Word Problem

A flying cannonball’s height is described by formula $y=−16t^2+300t$. Find the highest point of its trajectory. In how many seconds after the shot will cannonball be at the highest point? What is the ...
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1answer
26 views

Parabolas Inequalaties

If a vertical parabola opens upward, has its vertex in the third quadrant, and $y=ax^2+bx+c$ is the equation of this parabola, which of the following can be true? Sketch a curve for each possible ...
2
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1answer
22 views

Is there a term in mathematics for Metcalfe's Law?

Metcalfe's Law states that the value of a network is proportionate to the square of the number of users. This comes from the idea that there are $N*(N-1)/2$ pairs in a network of size $N$. Does this ...
1
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1answer
54 views

If $\alpha$ , $\beta$ are roots of the quadratic equation $x^2 -2p(x-4) -15 = 0$ , then answer the following .

What is the set of of values of $p$ for which one root is less than $1$ and the other is greater than $2$ ? A) $ (7/3,\infty) $ B) $ (-\infty,7/3) $ C) $ x \in R $ D) $ None $ Please tell me ...
3
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2answers
34 views

If $ ax^2 + 2bx + c = 0 $ and $ a_1x^2 + 2b_1x + c_1 = 0 $ have a common root , then prove the following. [closed]

If $a/a_1 , b/b_1 , c/c_1 $ are in A.P. then $ a_1 , b_1 , c_1 $ are in G.P. I have no idea , how to approach this . What I have thought : For the AP series $ a/a_1 = k - d $ the rest be k ...
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2answers
40 views

If $ 3x^2 + 2\alpha xy + 2y^2 + 2ax - 4y + 1 $ can be resolved into two linear factors, then prove the following.

Prove that : $ \alpha $ is a root of the equation $ x^2 + 4ax + 2a^2 + 6 = 0 $. What does it mean by "can be resolved into two linear factors"? If it means $( ax + b ) ( cx + d )$ , is it necessary ...
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2answers
117 views

$f(g(h(x)))=0$ has $8$ real roots

Find all quadratic polynomials $f(x),g(x)$ and $h(x)$ such that the polynomial $f(g(h(x)))=0$ has roots $1,2,3,4,5,6,7$ and $8$. I don't know what to do. Making a $8$ degree equation is quite ...
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1answer
15 views

Deriving Quadratic Function Using Table of Values

How do you derive a quadratic function given its table of values when there is no zero x value given? For example, x|1|2|3 y|-6|-3|4 I tried two methods in solving this but I can only get a. Any ...
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2answers
36 views

Nature of roots of $x^2+2(a-1)x+(a-5)=0$

A quadratic equation is given as $x^2+2(a-1)x+(a-5)=0$ then what could be the possible value of a if: a) The equation has positive roots b) The equation has roots of opposite sign ...
1
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1answer
36 views

Parabola conic section

Two tangents to the parabola $y^2= 8x$ meet the tangent at its vertex in the points $P$ and $Q$. If $|PQ| = 4$, prove that the locus of the point of the intersection of the two tangents is $y^2 = 8 ...
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1answer
27 views

Rewrite the equation $(x-a)^2 + (y-b)^2 = r^2$ to make $y$ a function of $x$

I'm trying hard to figure out how $(x-a)^2 + (y-b)^2 = r^2$ can be written as $y = b + \sqrt{r^2 - (x-a)^2}$. My book says that you’ll want to have $y$ as a function of $x$.
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1answer
40 views

Solving a system (3) of nonlinear equations

I want to solve the following euqations: $6y-2xz = 0 $ $6x-2yz = 0 $ $x^2 + y^2 -8 = 0 $ One way would be to multiply the first equations with y and the second with x and then substract the first ...
2
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0answers
27 views

How to use piecewise quadratic interpolation?

I'm attempting to get the hang of quadratic interpolation, in MatLab specifically, and I'm having trouble approaching the process of actually creating the spline equations. For example, I have 9 ...
4
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1answer
63 views

Change the variables in $Q(x,y,z)=(x-y+z-1)^2-2z+4$ to have $Q(f(u,v,w))=u^2+v$

I have a problem with this exercise. Initially, they gave me this polynom, and I had to complete the squares: $$Q(x,y,z)=x^2-2xy+2xz+y^2-2yz+z^2-2x+2y-4z+5.$$ I've done it, and I've checked with ...
3
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2answers
31 views

Condition for inverse of quadratic function - alternative solutions

I was helping my friend teacher to assemble a list of exercises to their precalculus students. So I came up with this problem: Let $f$ be a quadratic function, i.e. $$f(x) = ax^2 + bx + c,$$ ...
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2answers
54 views

Exponential simultaneous equations, short multiplication formulas - only for sneaky people

I am obligated to do some exercises from some Russian maths book and I solved them, but the teacher told us to use the smartest possible way to achieve this and I guess mine aren't sneaky enough and ...
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1answer
26 views

About quadric classification by completing square

I'm doing a seminar of geometry. We're learning how to classify quadrics with Maple, and there's a steps we have to follow in order to find what kind of quadric we have. Initially, they give me this: ...
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1answer
36 views

Number of real solutions of quadratic equation

I have the following question that puzzles me: How do I determine the number of non-trivial real solutions to the general equation $ax^2 + bxy + cy^2 = 0$ (up to a scalar)? My attempt was to fix $y ...
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3answers
47 views

How to solve a problem with a variable in both the base and exponent on opposite sides of an equation

I am working on systems of equations in Pre-Calculus, and I presented the teacher a question that I had been wondering for a while. $x^2 = 2^x$ The teacher couldn't figure it out after playing with ...
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4answers
32 views

How do you distribute this negative?

So I have $-(x - 2)^2$. Do I rewrite it as $-(x - 2)\cdot-(x - 2)$ and distribute the negative to the inside making it $(-x + 2)(-x + 2)$ or add the negative at the end of doing FOIL?
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3answers
42 views

What is the difference between the two real numbers that satisfy this equation?

What is the absolute difference between the two real numbers $x$ for which $(x+1)(x-1)(x-2) = (x+2)(x+3)(x-3)$? Express your answer in simplest radical form I tried guessing solutions but seeing ...
0
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2answers
48 views

Question on quadratic polynomials with real cofficients

Let$ P(x) = x^2 + ax + b$ be a quadratic polynomial with real coefficients. Suppose there are real number $s ≠ t$ such that $P(s) = t$ and $P(t) = s$. Prove that $ b - st$ is a root of the equation $ ...
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4answers
57 views

How do you factor $x^2-x-1$?

I know you can't have all integers, but how do you factor this anyway? Wolfram|Alpha gives me $-\frac{1}{4} (1+\sqrt{5}-2 x) (-1+\sqrt{5}+2 x)$. Cymath gives me ...
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1answer
12 views

Determining Parabola by Equation Problem

I need some help figuring out how to solve this problem. Which of the following could be a graph of the equation $ y= ax^2 + bx + c$ where $b^2 - 4ac = 0$ The picture below was the correct answer. ...
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1answer
15 views

Let $P(x)=(m^2+4m+5)x^2-4x+7,m\in R$.If $3\leq x\leq 5$,then find the minimum of the minimum value of $P(x).$

Let $P(x)=(m^2+4m+5)x^2-4x+7,m\in R$.If $3\leq x\leq 5$,then find the minimum of the minimum value of $P(x).$ The minimum value of $P(x)=(m^2+4m+5)x^2-4x+7$ occurs at $x=\frac{2}{m^2+4m+5}$ So the ...
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2answers
58 views

Sum of all values of $b$ if the difference between the largest and smallest values of the function $f(x)=x^2-2bx+1$ in the segment $[0,1]$ is $4$

Find sum of all possible values of the parameter $b$ if the difference between the largest and smallest values of the function $f(x)=x^2-2bx+1$ in the segment $[0,1]$ is $4$. I found that the ...
2
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1answer
43 views

How many solutions to $x^2-4x-\cos x=-8$?

$$x^2-4x-\cos x=-8$$ We want the number of solutions. Now I tried taking $\cos x$ as constant but by formula for root we get a trig equation which can't be solved. Any help? Thanks!
0
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4answers
88 views

If a quadratic polynomial $ax^2+bx+c$ has the form $(rx+s)^2$. Does it implies that $b^2-4ac=0$? [closed]

If a quadratic polynomial with integral coefficients $ax^2+bx+c$ is of the form $(rx+s)^2$ Can we say that the discriminant $$b^2-4ac=0?$$ If so how do you prove it?
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0answers
15 views

Linear Algebra and Quadratic Equations

I'm just wondering if Linear Algebra is concerned only with Linear equations? Can quadratic equations(or any higher power) also be considered under Linear Algebra? What does the term Linear stand for? ...
0
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1answer
16 views

I need help with variable expression

Good day! I have this coirdinate equation: $$\frac{gt^2}{2}+{v_y}t-\frac{5}{3}R=0$$ $$v=\sqrt{\frac{10}{9}*gR}$$ How i can express variable $t$ from this equation? I calculated this as quadratic ...
2
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0answers
56 views

Help solving the quadratic equation $ax^2-4bx+4bc-\frac{d^2}{a}=0$

I have been struggling to solve this quadratic equation in the variable $x$ with integral coefficients: $$ax^2-4bx+4bc-\frac{d^2}{a}=0$$ $a\neq 0$ of course.How do I ensure that $x$ is an integer? ...
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0answers
50 views

How many cows will eat the field?

Three pasture fields have areas of $\frac{10}{3}$, $10$ and $24$ acres, respectively. The fields initially are covered with grass of the same thickness and new grass grows on each at the same rate per ...
2
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2answers
41 views

If $a,b$ are the roots of the equation $x^2-2x+3=0$ obtain the equation whose roots are $a^3-3a^2+5a-2$, $b^3-b^2+b+5$

I have been trying this using sum of roots and product of roots but it gets too lengthy. So I found the roots of the given equation which are imaginary and tried to replace the values in the two given ...
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2answers
60 views

Prove Newtons method converges quadratically

f(x)= cosh(x) +cos(x) -3 Let x* be the none negative root of f. Prove that Newton's Method applied to f converges quadratically to x*. Really confused where to start for a proof. I understand that ...
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1answer
24 views

The number of integers $n$ such that the quadratic equation $nx^2+(n+1)x+(n+2)=0$ has rational roots is

The number of integers $n$ such that the quadratic equation $nx^2+(n+1)x+(n+2)=0$ has rational roots is $(A)0\hspace{1cm}(B)1\hspace{1cm}(C)2\hspace{1cm}(D)3$ The condition for the rational roots is ...
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0answers
16 views

solving quadratic equation in GF(2^m)

I am trying to implement Elliptic Curve Cryptography on software in GF(2^m). To do this, I need to be able to solve a quadratic equation, namely $x^2 + x = c$. After a lot of research, I know the ...
1
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1answer
83 views

Prove that $m-n$ divides $p$

Let $a_1,a_2,a_3$ be a non constant arithmetic Progression of integers with common difference $p$ and $b_1,b_2,b_3$ be a geometric Progression with common ratio $r$. Consider $3$ polynomials ...
0
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1answer
30 views

Quadratic Equation Application

Textbook exercise I get stuck "Amy is going to hold a concert at a stadium. The stadium can accommodate 12 000 people. If the price for each ticket is \$160, all the tickets will be sold. For every ...
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1answer
74 views

Finding a parametric form for the locus of points for a vanishing determinant

I need to find the locus of points in the real $(x, y)$ plane, in parametric form, satisfied by the equation \begin{equation}\det\begin{pmatrix} 0 & 1 & 1 & 1 & 1\\ 1 & 0 & ...
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1answer
20 views

Unique solution of quadratic expression

I have a quadratic expression: $x^HAx - B$ in which I have to solve for $x$. $A$ is a 4 x 4 positive definite hermitian matrix and $x$ is a 4 x 1 vector. I have solved it by considering Singular Value ...
0
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2answers
32 views

Factoring out an equation

I found this equation in a book: $$ m_0 v_0^2 + m_1 v_1^2 = m_0 v_{0_{Final}}^2 + m_1 v_{1_{Final}}^2.$$ It says that Notice that you have a different equation with the same two unknown ...
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1answer
16 views

How-To Quadratic Funcions and Graphs

I have the following problem: For $f(x) = −x^2 + 4x − 8$ the value of $-b\over{2a}$ is $2$. Find the $y$-coordinate of the vertex of the graph of this function. My book is severely lacking in ...
0
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1answer
38 views

Solving a quadratic expression

I have an equation of the form: $$x'Ax - B$$ where $A$ is a positive definite matrix. I want to solve this equation for $x$. Could anyone provide me with some suggestions on how to solve this kind of ...
0
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1answer
19 views

Minimizing a quadratic equation, with constraint.

I have a problem, I want to minimize this: $$ \min_{w} {w^\dagger H_1 w}\\ s.t. {w^\dagger H_2 w} = \mu^2 \\ ||w||^2 = 1 $$ with $\mu$ being real positive number, and $H_1, H_2$ are matrices with same ...
0
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1answer
17 views

Misstep With Discriminant and equations writable as quadratic form

I missed a step in my equation and would like to know what i'm doing wrong. I have the following equation: ${x^{4} - {\color{red}15} x^{2} + {\color{red}54} = 0}$ Now, we let ${y = x^2}$ We can ...