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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3answers
31 views

Vertex of the graph of a quadratic polynomial

This is what a website states: Before graphing a quadratic function we rearrange the equation, from this: $f(x) = ax^2 + bx + c$ To this: $f(x) = a(x-h)^2 + k$ Where: $h = -b/2a$ $k = ...
2
votes
2answers
198 views

Calculus approach to solve this Quadratic equation problem

Both roots of the equation $$(x-b) (x-c) +(x-a) (x-c) +(x-a) (x-b) = 0$$ are always positive , negative or real. Prove your result. By solving this equation I got $3x^2 - 2(a+b+c)x +ab + bc + ca = ...
4
votes
0answers
30 views

Complex roots of quartic polynomial

This is a question from an undergraduate course on Galois theory: Find all complex numbers which are roots of $P(T)=T^4+2T^2-\sqrt{6}T+\frac{3}{4}$ Can we use Galois theory to solves this? Or ...
3
votes
0answers
99 views

Suppose that a function $f(x)=ax^2+bx+c$, where $a,b,c$ are real constants, satisfy the relation..

Suppose that a function $f(x)=ax^2+bx+c$, where $a,b,c$ are real constants, satisfy the relation $$-1\leq f(x)\leq 1 $$ for all $-1\leq x\leq 1$, then the maximum value of $f'(x)$ is I think the ...
0
votes
2answers
22 views

How to solve specific parameters for a quadratic equation?

x^2+ax+a so that there are two different solutions x>5 First I set up that the discriminant is: D > 0 Then using Vieta's formula: a>25, a<10 But still, if I take 5 and 6 as solutions, I end ...
4
votes
2answers
231 views

Sum of cube roots of a quadratic

If $a$ and $b$ are the roots of $x^2 -5x + 8 = 0$. How do I find $\sqrt[3]{a} + \sqrt[3]{b}$ without finding the roots? I know how to evaluate $\sqrt[2]{a} + \sqrt[2]{b}$ by squaring and subbing for $...
0
votes
1answer
39 views

Solving a “simple” quadratic/quartic equation

Despite having solved quadratic quations for years I can't seem to be able to get the same result than maple on this one (not as simplified as Maple's), so I wonder if someone could not explain: I'm ...
0
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3answers
36 views

How to find the quadratic equation using 2 given solutions

Find the quadratic equation $ax^2 + bx + c = 0$, Such that $a=1$ and the solutions are: $3(\cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3})), 2(\cos(\frac{5\pi}{6}) + i\sin(\frac{5\pi}{6}))$
-1
votes
3answers
52 views

Solve without using quadratic formula: $\frac{4}{3x+3} = \frac{12}{x^2 - 1}$. [closed]

Solve without using quadratic formula: $\frac{4}{3x+3} = \frac{12}{x^2 - 1}$. Is there a way to solve this without using the quadratic formula? The quadratic formula is one of my biggest weaknesses, ...
0
votes
2answers
36 views

Solving exponential equation (quadratic type)

I fail trying to solve the following equation: $9^x-6^x-2^{2x+1}=0$ Trying to write it as a quadratic equation makes my constant term exponential $(3^x)^2-2^x3^x-2^{2x+1}=0$ How can I solve this ...
1
vote
0answers
30 views

$abc$-formula: $D >= 0$ or $D > 0$?

I don't really understand when to put $>$ or $<$ , $>=$ or $<=$ when working with the $abc$-formula. This is the $abc$-formula: We have a quadratic equation: $$ax^2+bx+c$$ Now for the $...
0
votes
1answer
15 views

Finding the equation of a quadratic with 2 points and a known slope. (SPLINES)

Sketch the spline of degree 2 with value 0.5 at x = 2.5 and the values 1, 1, 0, 0 at t0, . . . , t3, respectively. (t0=0, t1=2, t2=3, and t3=5) What is the value of the spline at x = 1 and 4? What I ...
0
votes
0answers
39 views

How to represent a system of quadratic equations in matrix form

Suppose I have two quadratic equation like the following: $2x^2 - 3x + 2$ $x^2 + 5x + 6$ I want to find the minimum values of these equation with the constraint that: $-3 \lt x \lt 5$ How ...
3
votes
2answers
47 views

If the quadratic equation $x^2+(2-\tan\theta)x-(1+\tan\theta)=0$ has two integral roots,

If the quadratic equation $x^2+(2-\tan\theta)x-(1+\tan\theta)=0$ has two integral roots,then find the sum of all possible values of $\theta$ in interval $(0,2\pi).$ The given quadratic equation is $...
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votes
1answer
46 views

Find the roots of the equation - $z^2 +12jz+64 = 0$

Just needing a little guidance. This is what I've done so far and I'm not sure if I'm doing it right. Using quadratic formula: $$z^2+12jz+64=0$$ $$ z= \frac{-12j ±\sqrt{(12j^2-4\times1\times64)}}{2\...
5
votes
2answers
89 views

All-Russian Olympiad question (composite of quadratics)

($1995$, All-Russian Olympiad, $9^{th}$ Graders, Final Round) Is it possible for the equation $f(g(h(x)))=0$, where $f, g$ and $h$ are quadratic functions, to have solutions $x=1,2,...,8$ ? I'm ...
-4
votes
7answers
126 views

Prove that $\frac{1-\sqrt{1-x^2}}{x}\le1$ [closed]

What are different ways to prove that: $$\frac{1-\sqrt{1-x^2}}{x}\le1$$ for $0<x<1$ Thanks!
3
votes
3answers
54 views

If $f(x)=Ax^2+Bx+C$ and $2A,A+B,C$ are integers, prove that $f(x)$ is integer whenever $x$ is an integer

If $f(x)=Ax^2+Bx+C$ and $2A,A+B,C$ are integers, prove that $f(x)$ is integer whenever $x$ is an integer. I found a way to prove the reverse statement. That is, I can prove that if $f(x)$ is integer ...
0
votes
6answers
51 views

Roots of quadratic equation are given by $b \pm \sqrt{b^2 - c}$

I was reading slides about the cancellation error in quadratic equations and it's written: The roots of the quadratic equation: $$x^2 - 2bx + c = 0$$ with $b^2 > c$ are given by $b \...
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votes
1answer
23 views

If $\alpha$ is the root(having least absolute value) of the equation$x^2-ax-1=0(a\in R^+)$

If $\alpha$ is the root(having least absolute value) of the equation$x^2-ax-1=0(a\in R^+)$,then which of the following relation is correct? $(A)\alpha<-1\hspace{1cm}(B)-1<\alpha<0\hspace{1cm}(...
0
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0answers
16 views

How To Estimate Parameters In A Linear Regression Graph?

I'm in the beginning of my machine learning course and I'm stuck at a quiz. A set of questions are setup like the following... Linear Regression Graph (Quadratic) Possible answers are (Select all ...
0
votes
1answer
47 views

Convex optimization: Piece-wise, quadratic objective

This question is about convex optimization with a convex objective function, which is defined piecewise. We have two functions, a concave function A(x) and a strictly convex, increasing function B(x), ...
1
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2answers
169 views

$a,b,c$ are the sides and $A,B,C$ are the angles of a triangle. If the roots of the equation $a(b-c)x^2+b(c-a)x+c(a-b)=0$ are equal then,

$a,b,c$ are the sides of a $\triangle ABC$ and $A,B,C$ are the respective angles. If the roots of the equation $a(b-c)x^2+b(c-a)x+c(a-b)=0$ are equal then $\sin^2 \bigl(\frac{A}{2}\bigr), \sin^2 \bigl(...
0
votes
0answers
15 views

Value of Taylor Approximation

I'm studying Trust Region Methods and there's one simple thing I'm finding it hard to wrap my head around. The premise of the method is to approximate $f(x)$ as a quadratic model: $$m_n(x)=f(x_n)+\...
13
votes
1answer
316 views

Convergence of the quadratic map $\left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2$?

Edit - I changed the title and much of the body to better reflect my full question. The old one I don't really care about, although I appreciate Fabian's answer of course. Here is the plot for the ...
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votes
2answers
336 views

solving $y=4/x+\sqrt{x+0.2−5x}$ [closed]

it's actually $y=\frac{4}{x}+\sqrt{x+0.2-5x}$ (see algebra problem) $$y=\frac{4}{x}+\sqrt{x+0.2-5x}$$ if $x=\frac45$ what is y?
2
votes
1answer
48 views

Weird property of quadratic equations

Today, we learnt about the quadratic formula, and I noticed a strange property of (seemingly) all quadratic equations. If we call the two solutions of any arbitrary quadratic equation $x_1$ and $x_2$: ...
4
votes
3answers
48 views

Possible number of terms in an Arithmetic Progression

The sum of the first $n$ $(n>1)$ terms of the A.P. is $153$ and the common difference is $2$. If the first term is an integer , then number of possible values of $n$ is $a)$ $3$ $b)$ $4$ $c)$ $...
1
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0answers
35 views

How to constrain the linear least squares fit of a quadratic polynomial with known constraints

How to constrain this fit I have some function , $f(x) = a x^2+b x+c$ , with the constraints $a<0$ and $c = \frac{b^2}{4a}+\frac{1}{2}ln(\frac{-a}{\pi})$ I have measured $f(x)$ for some $x$. Can ...
1
vote
4answers
554 views

How do the roots change

How do the roots of the quadratic equation $ax^2+bx+c=0$ change when $b$ and $c$ retain constant values and $a$ tends to zero? ($b\neq 0$)
0
votes
2answers
51 views

What is the solution set of $ \displaystyle\log _x\left({x+5\over 1-3x}\right)>0$?

What is the solution set of $ \displaystyle\log _x\left({x+5\over 1-3x}\right)>0$? I'm getting $0 < x < 1/3$ but that is wrong answer.
1
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0answers
29 views

Solve quadratic vector equation, with variable hidden inside scalar

Let $\vec{f}$ $m\times 1$ unknown vector, given $n\times 1$ vector $\vec{F}$, $n\times m$ matrix A ($n<m$), nonzero vector $\vec{v}$ from the nullspace of A ($Av=0$), non-invertible symmetric ...
-1
votes
2answers
66 views

Correct proof of a theorm [closed]

Hi Can someone please give me an idea where to start with this Thanks Steve
2
votes
1answer
44 views

What curve is traced out by the vertex of the parabola $ax^2+bx+c$ as $b$ varies?

Consider the parabola $\rho$ given be the equation $y = ax^2 + bx + c$. Recall that varying $a$ in this equation stretches/squashes $\rho$ and that varying $c$ shifts $\rho$ vertically. The change in ...
0
votes
1answer
31 views

How to find the integral value of $a$ for which $f(x) = x^2 - 6ax + 3 - 2a + 9a^2$ is surjective

Let $f:\mathbb{R} \to [1, \infty)$ be defined by $f(x)=x^2-6ax+3-2a+9a^2$. The integral value of $a$ for which $f(x)$ is surjective is equal to I tried putting $f(x)=1$. Is this the right approach?
1
vote
1answer
27 views

Composition of three quadratic functions

Is it possible to find three quadratic functions $f(x),g(x)$ and $h(x)$ such that $f(g(h(x)))$ has $-6,-5,-4,-2,1,3,4,5$ as its roots? I understand that the composition of three quadratic functions ...
0
votes
1answer
31 views

Find $f(\mathbb D)$

If $f(x)=\frac{\sqrt{x}-x}{\sqrt{x}+2}$ and $x\in\mathbb D=[0,\infty]$ find $$f(\mathbb D)$$. I've tried to solve equation $y=f(x)$ and stopped to $x+(y-1)\sqrt{x}+2y=0$.
1
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3answers
42 views

Why does the width of the graph of a parabola depend only on $a$, not $b$?

Lets assume a quadratic function $y = ax^2 + bx +c$. My book says how wide or narrow the graph is depends on the size of $|a|$ My question is why doesn't it depend on $b$ also? If you, say, increase ...
4
votes
3answers
100 views

Find $a^{100}+b^{100}+ab$

$a$ and $b$ are the roots of the equation $x^2+x+1=0$. Then what is the value of $a^{100}+b^{100}+ab$? Here's what I found out: $$a+b=-1$$ $$ab=1$$ but how to use this to find that I don't know! ...
3
votes
1answer
89 views

Is there a number besides $\phi$ that either squared or added one gives the same answer?

Those who know golden ratio $\phi$ (phi) constant, know for sure that it is an interesting constant. It is roughly $\phi=1.618034...$ . It is present almost everywhere in nature and it has many very ...
6
votes
8answers
953 views

Solving the following trigonometric equation: $\sin x + \cos x = \frac{1}{3} $

I have to solve the following equation: $$\sin x + \cos x = \dfrac{1}{3} $$ I use the following substitution: $$\sin^2 x + \cos^2 x = 1 \longrightarrow \sin x = \sqrt{1-\cos^2 x}$$ And by ...
2
votes
1answer
22 views

How to determine the number of free variables in a nonlinear system?

Consider the system of equations $$ \begin{align} a_1 b_2 &= c_1 \\ a_1 b_3 &= c_2 \\ a_2 b_1 &= c_3 \\ a_2 b_3 &= c_4 \\ a_3 b_1 &= c_5 \\ a_3 b_2 &= c_6 \\ \end{align} $$ ...
0
votes
2answers
23 views

Invertability of quadratic function

There is a proof in my lin alg book that shows that if T is a self adjoint operator, $T^2+Tb+c$ is invertible when $b^2 < 4c$?. I understand this proof. Why is it that the above holds, and yet the ...
0
votes
2answers
52 views

Simply factoring a quadratic equation

On pp 255 - 256 (footnote 7) of "Love & Math", Edward Frenkel states that we can factor a quadratic in terms of its solutions $x_1$ and $x_2$ as: $ax^2 + bx + c = a(x - x_1)(x - x_2)$ Where does ...
1
vote
2answers
52 views

Roots of a quadratic equation

If $ a $ , $b$ are the roots of $ x^2+x+1=0 $ , then the equation whose roots are $a^k $ and $b ^k$ where k is a positive integer not divisible by 3 is $a)$ $x^2 - x + 1 = 0$ $b)$ $x^2 + x+1 = 0$ $...
5
votes
2answers
70 views

What is the range of $f :R → R$, and $f(x) = x^2 + 6x − 8$

I have this discrete math question I have done completing the square but not sure how to continue. May I get some guide? Thanks! What is the range of $f :R → R$, and $f(x) = x^2 + 6x − 8$ $f(x)=x^2+...
1
vote
1answer
60 views

Finding integers satisfying an equation.

Suppose $p$ is a prime greater than $3$. Find all pairs of integers $(a,b)$ satisfying the equation $$a^2+3ab+2p(a+b)+p^2=0$$ A good way to start (probably) was to complete the square, $$p^2+2p(a+b)...
3
votes
6answers
165 views

Find the roots of quadratic polynomial given one root of another quadratic polynomial?

if $a,b,c$ are Real numbers and $1$ is a root of equation $ax^2+bx+c=0$ then curve $y = 4ax^2+3bx+2c$ , (a is not zero) intersects $x$ axis at how many points? I get a relation $a+b+c = 0$ I tried ...
0
votes
1answer
34 views

Determining the number of roots

Given a set of two equations (one linear and one quadratic in $x$ and $y$) as follows:- $$ax + by + c = 0 \tag 1$$ $$Ax^2 + Bxy +Cy^2 + Dx + Ey + F = 0 \tag 2$$ What are the conditions that can be ...
1
vote
0answers
45 views

Taylor's formula and its quadratic term

I struggle with the following problem: For a function $$f: \mathbb{C} \rightarrow \mathbb{R}~,$$ $f$ attains its maximum for $z_0= e^{i\pi/3}$, $f(z_0)=F_{max}.$ Assume we may use Taylor's theorem (...