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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2answers
54 views

How to simplify a fraction

I'm having some difficulty understanding how the expression $$x = \frac{12 \pm 4\sqrt{7}}{8}$$ is simplified to $$x = \frac{3 \pm \sqrt{7}}{2}$$ Where does the $4$ go? It just disappears?
1
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2answers
47 views

Assume this equation has distinct roots. Prove $k = -1/2$ without using Vieta's formulas.

Given $(1-2k)x^2 - (3k+4)x + 2 = 0$ for some $k \in \mathbb{R}\setminus\{1/2\}$, suppose $x_1$ and $x_2$ are distinct roots of the equation such that $x_1 x_2 = 1$. Without using Vieta's formulas, ...
0
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0answers
21 views

Under which conditions do positive real roots exist in a quadratic of two variables?

Suppose we have a quadratic polynomial equation in two variables: $$p(x,y) = ax^2 + by^2 + cxy + dx + ey + f = 0$$ Under which conditions does at least one solution $p(x,y)=0$ exist with $x$ and $y$ ...
0
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1answer
65 views

When are we allowed to match coefficients?

Related to this answer: Find k in $(1−2k)x^2−(3k+4)x+2=0$ given facts about the roots. In the partial fraction decomposition of $\frac{1}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1}$, we have: $0x + ...
2
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4answers
64 views

Why does the coefficient tell the way a quadratic parabola “opens”?

I've seen a lot of texts referring to the coefficients and their sign for determining which way a parabola opens up. But is there more than this kind of "thumb rule" to it? That is, how was it proven ...
2
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2answers
34 views

Diophantine Equation or Ellipse

I guess this is something to do with circle The question is: "Given x and y are real numbers, such that $2x^2 + 3y^2 - 4x - 12y = -14$, find xy." What is the trick ?
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2answers
78 views

Find k in $(1-2k)x^2 - (3k+4)x + 2 = 0$ given facts about the roots.

The exact instruction in my book is: A quadratic equation $(1-2k)x^2 - (3k+4)x + 2 = 0$ is given. Find the value of k for each of the following conditions. (I got 46a and 46b) (c) A ...
0
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1answer
21 views

Trouble solving a quadratic equality in x which has a single unknown

I have a relatively straightforward inequality to solve but I am getting stuck because I don't know what to do with the unknown... $$x^2 + b^2 > 0, b > 0$$ The aim is to solve the inequality ...
0
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1answer
31 views

$x^{\frac{1}{2}}-3x^{\frac{1}{4}}-10=0$

$$x^{\frac{1}{2}}-3x^{\frac{1}{4}}-10=0$$ $$t=x^{\frac{1}{4}}$$ $$t^2-3t-10=0$$ $$\Delta=49$$ $$t_1=5$$ $$t_2=-2$$ And now I'm not sure what should I do... $$x_1^{\frac{1}{4}}=5$$ $$x_1=5^4=625$$ I'm ...
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2answers
31 views

the result after exploding second degree equation

Why is $$(1+x+x^2)^2 = 1+x^2+(x^2)^2+2(x+x^2+x^3)?$$ My professor wrote that in the class but he didn't explain why. Thank you
3
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5answers
568 views

Solving Quartic Equations

Given the following quartic equation: $$x^4-2x^3-7x^2+8x+12=0$$ Could anyone give some techniques required to solve any quartic equation (apart from this one) if they exist?
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1answer
27 views

Find the values of k in a quadratic equation

I have to find the values of k such that this quadratic equation $(k-5)x^2-4kx+k-2=0$ allows real and negative solutions of x. Can you help me?
11
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0answers
106 views

On the prime-generating polynomial $m^2+m+234505015943235329417$

In 2009, J. Waldvogel and Peter Leikauf found the remarkable Euler-like polynomial, $$F(m)=m^2+m+234505015943235329417$$ which is prime for $m=0\to20$, but composite for $m=21$. Define, ...
1
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2answers
45 views

Prove that $f_n = \frac{\alpha^n - \beta^n}{\alpha - \beta}$ for all $n \in \mathbb Z^+$.

Let $\alpha = \frac{1+\sqrt{5}}{2} \hskip 20pt \beta = \frac{1-\sqrt{5}}{2}$ be the two real roots of the quadratic equation $x^2 - x - 1 = 0.$ Prove that $f_n = \frac{\alpha^n - \beta^n}{\alpha - ...
0
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1answer
52 views

Why 3 root 2 is equal to 3 divided by root 2 in this quadratic equation

As part of my research into quadratics, I am trying to show algebraically the following equation : $${x^2-3\sqrt2 x + 4} = 0$$ alternatively shown as $x^2 - 3 \cdot 2^{\frac{1}{2}}x + 4 = 0$ Now, ...
0
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4answers
64 views

Find inverse of $7x^{2}-112x+448$

Given the function $\; f(x) = 7x^{2}-112x+448, \;$ for $x\ge 8, \;$ find $\displaystyle \;$ $f^{-1}(x)$. To find inverse, I should just solve for x in terms of y: $$y = 7x^{2}-112x+448$$ I can ...
1
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0answers
24 views

discriminant of a quadratic function

Let $f$ and $g$ be a quadratic funtcions. Assume that $|f(x)|\geq |g(x)|$ for all $x\in\mathbb{R}$. How to show that $|d_f|\geq|d_g|$, where $d_h$ denote the discriminant of an arbitrary quadratic ...
0
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2answers
34 views

Quadratic equation with indicies as another quadratic.

I'm trying to solve the following question below (Please do excuse the formatting)... $$x^{x^2-7x+11} = 1$$ Now, so far, I have calculated that as $1 =x^0$ that I can form an equation which is ...
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2answers
30 views

Deducing the focus of a parabolic mirror

I've been sitting on it for past three hours but I am unable to spot anything interesting. The first part was obvious but I can't answer the second question (b ii). How can I proof this statement k=a? ...
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1answer
38 views

Is this a second grade system?

There are systems that are obviously of second grade, such as: $$x^2 + y = 1$$ $$y = x - 1$$ The definition of an n-th grade system is: The grade of a system is the product of the grade of all ...
2
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0answers
69 views

Parabola Terminology

In Danish we call the two halves of a parabola that goes out to each side from the vertex branches like branches on a tree. Is there a name for them in English? Are they just called halves or maybe ...
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1answer
43 views

please solve this question on quadratic equations

Consider the quadratic function for $x$: $$ y=x^2-8px+16p^2+p+6 $$ Where $p$ is a real number. The vertex of the parabola described by this function lies in the second quadrant if ...
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2answers
107 views
0
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3answers
41 views

The quadratic equation $x^2 + Lx + M = 0$

The question: The quadratic equation $ x^2 + Lx + M = 0$ has one root twice the other. a) Prove that the roots are rational whenever L is rational. I was able to find out that due to one root ...
0
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1answer
29 views

Quadratic possibilities: how to solve this?

Given: $2x^2+kx+2$, under what conditions is it positive definite? Since it's a quadratic, we take the discriminant $\Delta = k^2 - 16$. For it to be definitive, we need $\Delta \geq 0 $, so we ...
0
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4answers
65 views

For which $x \in \mathbb{R}$ is the inequality $x^2-5x - 6\ge 0$ true?

The inequality is: $$x^{2}-5x - 6 \geq 0. $$ I know this sounds like a stupid question, but via just guessing and checking I got, that $$3 \leq x \leq 0$$ but I am unsure how to formulate a proper ...
0
votes
1answer
27 views

Write a quadratic equation in the form y=Ax^2 + Bx +C?

the question is to turn this question {1/4} double root to standard form. What i got is x = 1/4, x= 1/4 once you multiply them you get x^2 -1/2x -1/16 I think not sure if I'm correct and thanks to ...
0
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1answer
59 views

Solve using Quadratic Formula Question, What went wrong?

Okay the question is to solve using quadratic formula and the question is $10r^2-6=0$ and after using the $-b \pm \frac{\sqrt{ b^2-4ac}}{2a}$ I got $-0 \pm 0^2-\frac{\sqrt{240}}{20}$...and I don't ...
0
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4answers
62 views

Inequation with 2 variables - how to proceed?

Find the highest value of $K$ such that $$x^2 - 10x + 40 \ge K$$ $ \forall x \in R$. Step 1: I'll assume $K$ is equal to $0$ just to make it simpler (Is it OK to do so?) $$x^2 - 10x + 40 \ge ...
0
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0answers
23 views

Normal forms of a quadratic form of two variables.

If we are given a form $Q(x,y) = ax^2 + 2bxy +cy^2$, then, using a rotation (or a linear change in coordinates) we may eliminate the $xy$ cross term to obtain the following form: $Q(X,Y) = AX^2 + ...
1
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0answers
42 views

Help solving this 2nd Order Differential Equation

I am wondering how I finish off the solution to the following 2nd Order differential equation: $$3u_{tt} + 2u_{xt} - 8u_{xx} = \cos(x + t)$$ I know that I have to factorize it and get the following: ...
2
votes
1answer
47 views

Finding the constant for a quadratic. Two methods; which one is correct and why?

The question reads $kx^2 + (k+2)x - 3 = 0$ has roots which are real and positive. Find the possible values k might have. Now, since it has real and positive roots, the discriminant $\Delta{d} > ...
2
votes
3answers
57 views

Quadratic Inequalities: My answer is different from the ones provided.

The problem is: $$x^2 + 2x - 1 < 0$$ Step 1: Move the $1$ to the other side. $$x^2 + 2x < 1$$ Step 2: Add $1$ to both sides to complete the quadratic equation. $$x^2 + 2x + 1 < 2$$ ...
0
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2answers
37 views

Vertical length of two roots

Let $f(x) = x^3 + 3bx^2 + 3cx + d$. (a) Show that $y = f(x)$ has two distinct turning points if and only if $b^2 > c$. (b) If $b^2 > c$, show that the vertical distance between the turning ...
2
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2answers
54 views

Can I find a better solution of $\sin(x^2 + x) = \frac 1 2$ for $x$?

It's obvious that $$x^2 + x = (-1)^k \arcsin \frac 1 2 + \pi k$$ where $k \in \Bbb Z$. If I treat $(-1)^k \arcsin \frac 1 2 + \pi k$ as a constant, I get the answer: $$x_{1, 2} = \frac {-1 \pm ...
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2answers
31 views

simultaneous equation help

Can someone help me with to solve this system of equations ? $$ \left\{ \begin{array}{c} y=x+1 \\ y^2+2x^2=2 \end{array} \right. $$
6
votes
2answers
90 views

$a,b,c\in \Bbb Z$ and $a\cdot b\cdot c$ is a root of $ax^2+bx+c$.

I was curious if there are quadratic equations where $a,b,c\in \Bbb Z$ and $a\cdot b\cdot c$ is a root of $ax^2+bx+c$. So trivially if $c=0$, $a$ and $b$ can be arbitrary, and if either $a$ or $b$ is ...
2
votes
2answers
35 views

What is the meaning of a discriminant graphically?

I know that if $b^2-4ac>0$, then there are real solutions and so on. But for any quadratic function, where is the discriminant present in the plot? For example, $x^2-8x+7=f(x)$. Now how is ...
2
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2answers
64 views

Firing Solution on a Moving Target

I need to calculate the 3-component $\vec V$, which is the gun barrel vector needed to hit a target moving at a constant velocity. To find this information I'll also need to find $t$ which is the time ...
0
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6answers
100 views

Find solution to complex equation $iz^2+(3-i)z-(1+2i)=0$

Find all the complex solutions to the equation $$iz^2+(3-i)z-(1+2i)=0$$ I've tried to solve this equation with two different approaches but in both cases I couldn't arrive to anything. 1) If ...
0
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1answer
31 views

How should I proceed with this inequation?

$$(2x + 1)(3x - 5) - 4x(x-3) + 7 \leq 0$$ (is less or equal than "0", someone help me to edit that part). Step 1: $$6x^2 - 10x + 3x - 5 - 4x^2 + 12x + 7 \leq 0$$ Step 2: $$2x^2 - 19x + 7 \leq 0$$ ...
2
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2answers
24 views

Finding A Quadratic Whose Roots Equals Intercept On Axes and Area Equals A

How to find the quadratic equation whose roots are the x and y intercepts of the line passing through $(1,1)$ and making a triangle of area A with the axes? Ok I'm getting $(1-m)(1-1/m)=A$ and ...
0
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1answer
73 views

Solve the quadratic equation $(2-y)^4=3(2-y)^2+1$

Solve $$(2-y)^4=3(2-y)^2+1$$ The answer is supposed to be $y=4\pm \sqrt{6+\frac{13}2}$. I have tried to work this problem out but I cannot get the answer that is in the book.
0
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3answers
33 views

Finding remainder

Okay I saw this one on a test so here it goes: A polynomial of (degree > 3) when divided by $ (x-1)^2$ and $x-3$ leaves the remainder $2x+1$ and $15$ respectively. The remainder when it is divided by ...
8
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4answers
763 views

Simple Trig Equations - Why is it Wrong to Cancel Trig Terms?

In the following problem, I first did it using a cancellation of $sin^2\theta$, working shown below, which gave the wrong answer. Having looked at the question again, I saw it could be solved by ...
0
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4answers
79 views

Highschool Algebra: $n^2 = 18n$?

I'm beginning to get into maths outside of school and at the moment I'm refreshing myself on the basics which explains why this question appears to be so simple. I formulated this equation to find ...
4
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2answers
135 views

How to solve this system of nonlinear equations?

How to solve these equations for $a$, $b$, $c$ and $x$? I have the following: \begin{align} 1 &= 2a+b+c\\ a &= (a+b)x + 0.25(a+c)\\ a&=(a+c)(1-x)\\ b&=a(1-x)+c(x-0.25)\\ ...
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2answers
25 views

Need help with tangents to a quadratic

The quadratic $y=kx^2+(3k-1)x-1$ and the straight line $y=(k+1)x-11$ meet. Find the range of value(s) of $k$ such that the line is a tangent to the curve. Got this question for school. Seems really ...
5
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3answers
108 views

Range of a Rational Function

How to find the Range of function $$f(x)= \frac{x^2-3x-4}{x^2 - 3x +4}$$ I tried to equate the expression to $y$, then cross multiplied $$ y= \frac{x^2-3x-4}{x^2 - 3x +4}$$ $$ y(x^2 - 3x +4)= ...
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0answers
35 views

Find first positive perfect square in polynomial time

I have a quadratic. for example $$1x^2+6884x+3297$$ Is it possible to find the first perfect square in the series in polynomial time where both x and y are whole positive integers. In the above ...