# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ($b\neq 0$)
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### 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.
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 $x+(y-1)\sqrt{x}+2y=0$.
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### 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 ...
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### 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! ...
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### 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 ...
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### 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 ...
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### 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} ...
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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 ...
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### 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 ...