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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31 views

Solving octic equation using quadratic formula

According to the wikipedia article on octic equations, octic equations of the form $ax^8 \pm bx^4 \pm c = 0$ can be solved using the quadratic formula. How might one actually do this?
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0answers
41 views

question on quadratic expansion

I have been trying to solve this question, but no luck so far, any help would be appreciated. Let $a,b,c > 0$ be such that $a^2 + b^2 -2bc =100, \ 2ab -c^2 = 100$. Then the value of ...
3
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4answers
67 views

$\sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 = 38$ - I keep getting imaginary numbers

$$\sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 = 38$$ My attempt $\sqrt{x+938^2} + \sqrt{x + 140^2} = 1116$ $(\sqrt{x+938^2} + \sqrt{x + 140^2})^2 = (1116)^2$ $x+938^2 + ...
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1answer
16 views

find the equation of the diameter which passes through the origin.

I am given the equation of the circle $x^2+y^2−4x+6y=14$, and I am told to find the equation of the diameter which passes through the origin. However, I am unsure as to how to do this.
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2answers
19 views

Quadratic inequality (Sign Reversal?)

I have the following inequality $\ (2x-3)^2-9>7$ I can reduce it down to $\ 2x-3>±4$ Now here is where I encounter a problem. Apparently the next step is $\ 2x-3>4 ~OR~ 2x-3<-4 $ ...
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1answer
25 views

Value of $a$ such that range contains the interval $[0,1]$

Find the number of integral values of $a$ in the interval $[0,100]$ so that the range of the function $y= \frac{x+a}{x^2-1}$, $x\in R$ contains the interval $[0,1]$? After rearranging $y= ...
3
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1answer
88 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 ...
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0answers
14 views

Approximating a quadratic term in the constraint set as 2nd order Taylor expansion

I have an optimization problem in the following form: $$\min_{x,y} f(x)+g(y)$$ $$s.t.$$ $$Ax+h(y)=0$$ where $h(y)$ is a quadratic in $y$. Instead of solving this problem directly, $h(y)$ is ...
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2answers
24 views

Value of qudaratic equation

In my exams, I was asked to calculate value of Quadratic Equation from given value of a, b, ...
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1answer
28 views

Solving a cubic function with P and Q

I have been struggling a little bit over solving cubic functions. I have been trying to use the P and Q method. So the question is What is the approximate value of the greatest zero of $f(x) = x^3 - ...
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0answers
8 views

Vertex form for inverse equations

I was wondering how to use interpret an inverse equation into vertex form, or y=a(x-h)+k So I have this problem: Problem and I used 1/4c and found the vertex (1/2,0) to determine that a = 1/14 and h ...
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1answer
35 views

$x\in\overline{F}$ is in $F(\sqrt{F})$ $\iff$ $F\subset F(x)$ is finite Galois extension with Gal$(F(x)/F)$ abelian of exponent $2$

Let $F$ be a field of characteristic that is not $2$. I want to prove that $x\in\overline{F}$ is in $F(\sqrt{F})$ $\iff$ $F\subset F(x)$ is a finite Galois extension for which the group ...
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1answer
28 views

Second Order Accurate Interpolation

On a grid I am having the values of a physical quantity say for example Temperature, at the E,W,N,S and P node all of them being calculated using a second order discretization scheme. I want a second ...
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2answers
20 views

quadratic equation form maximum solutions

My Pearson intermediate algebra book has a "concept check" question in its section on solving equations by using quadratic methods. These questions are supposed to highlight fundamental concepts that ...
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3answers
28 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 = ...
3
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1answer
59 views

Why does $\left(\frac b2\right)^2$ "geometrically complete the square?

I was just reading this MathisFun article on completing the square. It states that geometry can help complete the square. It starts off with a square and a rectangle (pictures come from link): ...
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2answers
193 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 ...
3
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1answer
92 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 ...
2
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2answers
718 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
28 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 ...
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6answers
47 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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2answers
20 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 ...
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2answers
108 views

How can we prove that a quadratic equation has at most 2 roots?

A quad equation can be factored into two factors containing $x $, but how can we prove that there no other sets of different factors yielding OTHER VALUES OF $X $?
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1answer
481 views

How do I prove a quadratic is always positive or negative for x?

I looked this up and seen something that was beyond my A-Level Maths course. In class we are doing the discriminant and sketching quadratic graphs, so it is nothing advanced. My teacher completed ...
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2answers
225 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 ...
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1answer
35 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 ...
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1answer
467 views

Find pressure in a sinusoidal function

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 ...
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3answers
33 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: $3(\cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3})), 2(\cos(\frac{5\pi}{6}) + i\sin(\frac{5\pi}{6}))$
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2answers
98 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 ...
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3answers
50 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, ...
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2answers
30 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 ...
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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 ...
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1answer
13 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 ...
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0answers
28 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 ...
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2answers
44 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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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 ...
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1answer
281 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 ...
2
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1answer
127 views

Proving the minimum value of (x+a)(x+b)/(x+c)

Show that the minimum value of $\frac {(x+a)(x+b)}{(x+c)}$, where a$\gt$c, b$\gt$c, is $(\sqrt{a-c}+\sqrt{b-c})^{2}$ for real values of x$\gt-c$. I did $$\frac {(x+a)(x+b)}{(x+c)}=y$$ and then took ...
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2answers
77 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 ...
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7answers
118 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!
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1answer
1k views

Difference and Quotient of roots of a quadratic equation

In school we are taught the sum and product of roots of $y= ax^2+bx+c$. But are not the difference and quotient of roots equally important? Difference $= \dfrac{\sqrt{b^2-4ac}}{a}$ and Quotient $ ...
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1answer
26 views

Solving the following equation by factorisation

The given equation is, $\frac{m}{n}x^2+\frac{n}{m}=1-2x$ What I've tried, Multiplying the equation by $n$, we get $mx^2+\frac{n^2}{m}x=n-2nx$ Now what? I am completely confused about what to do. ...
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3answers
51 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 ...
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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? ...
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0answers
13 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 ...
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1answer
33 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), ...
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3answers
39 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 $a)$ $3$ $b)$ $4$ $c)$ ...
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0answers
33 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 ...
0
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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: ...
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2answers
73 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?