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
15 views

Graphs and Quadratic Equations?

Question: A ball is dropped from a height of $18$ metres. The quadratic equation $h = -2t^2 + 18$ gives you the height, $h$ metres, of the ball, after $t$ seconds. After how many seconds does the ...
2
votes
2answers
58 views

Show that $2\cosh z + \sinh z = i$

The equation is $$2\cosh z + \sinh z = i$$ I used the following formulas: $$\cosh z = \frac{e^z + e^{-z}}{2}, \sinh z = \frac{e^z - e^{-z}}{2}$$ to reduce this equation to $$3e^z - e^{-z} = 2i$$ but ...
-2
votes
2answers
41 views

The equation $x^2+ax+(b+2)=0$ has real roots. Find the minimum value of $a^2+b^2$. [closed]

The equation $x^2+ax+(b+2)=0$ has real roots. Find the minimum value of $a^2+b^2$.
1
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1answer
47 views

Center and axis of Quadratic Surface

Given a Quadratic Surface in the form: $ax^2+by^2+cz^2+2fyz+2gzx+2hxy+2px+2qy+2rz+d=0$ I know how to decide which kind of surface is represented (http://mathworld.wolfram.com/QuadraticSurface.html). ...
2
votes
1answer
29 views

quadratic pre colloge problem

If c is a real number and the negative of one of the solutions of $x^2 -3x +c=0$ is a solution of $x^2 +3x -c=0$ then the solution of $x^2-3x+c=0$ are.... i cant do this problem due to language ...
0
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2answers
38 views

Solving a logarithmic equation where the logarhitm is exponentiated

I have troubles solving the following logarthitmic equation. $$ \ 2(\log_x{\sqrt7})^2-\log_x{\sqrt7}-1 =0 $$ The results are supposed to be $ \ x_1 = {\frac{1}{7}}, x_2 = \sqrt7 $ I have tried ...
0
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5answers
65 views

How to solve quadratic equation in 4 variables?

I am trying to solve the equation $a^2 + b^2 + c^2 + d^2 = \alpha$ where alpha is a constant and $a, b, c$, and $d$ are the $4$ variables. I need to find some integer values of $a, b, c$ and $d$ which ...
1
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1answer
18 views

Finding the intersection between a plane and an ellipsoid

I have an ellipsoid (That can be rotated), described by its conic equation : $$E : a_1x² + a_2y² + a_3z² + 2a_4xy + 2a_5xz + 2a_6yz + a_7x + a_8y + a_9z + \gamma = 0$$ This ellipsoid can be ...
0
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1answer
47 views

$p$ and $q$ be the roots of the polynomial $mx^2 + x(2-m) + 3$. [closed]

Let $p$ and $q$ be the roots of the polynomial $mx^2 + x(2-m) + 3$. Let $m_1$ ,$m_2$ be two values of m satisfying $\frac{p}{q} + \frac{q}{p}= \frac{2}{3}$. determine numerical value of $m_1/(m_2)^2+...
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3answers
46 views

Factoring $12e^{2x} - 32e^x + 16$ [closed]

Can you help me solve the quadratic equation $12e^{2x}-32e^x+16$ by factoring please?
2
votes
1answer
28 views

If $a =\cos (\frac{2 \pi}{7})+i \sin (\frac{2 \pi}{7}) $ then construct a quadratic equation.

If $a =\cos (\frac{2 \pi}{7})+i \sin (\frac{2 \pi}{7}) $, then find a quadratic equation whose roots are $\alpha = a + a^2 + a^4$ and $ \beta = a^3 + a^5 + a^7$ . Using the fact that sum of $7th$ ...
1
vote
3answers
180 views

Where am i going wrong in solving this equation?

Fing the least value of $a$ for which $f(x)$ is increasing, where $$f(x)=2e^x-ae^{-x}+(2a+1)x-3$$ What i tried for increasing $f'(x)\ge 0, \forall x\in \mathbb R$. So $$f'(x)=2e^x+ae^{-x}+2a+1\...
0
votes
2answers
32 views

Graphing log with number in front of “log”

When I have something like $y = log_2(x)$ I know that I have to turn it into exponential form and get: $2^y = x$. Next, I make a table for $X,Y$ and choose about 5 values for $y$, typically $-1, 0, 1, ...
1
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0answers
19 views

Simple Quadratic Question Word Problem

On a site, I read a word problem: A company charges 1.25 per ride and currently averages 10000 riders per day. The company needs to increase revenue, but found that each $.10 increase in fare results ...
0
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1answer
27 views

Cone under similarity transformation

Suppose we have a cone passing through the origin of $xyz$ coordinate system. Now, the question is that whether we can find an invertible transformation on this coordinate system that turns the cone ...
2
votes
2answers
46 views

Quadratic formula errors

I'm clearly making a silly mistake here, but I can't see it. EDIT: I missed brackets when typing out the expression to calculate. Apologies for timewasting. I have the equation $(2x + 3)(5x + 1)=0$. ...
1
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2answers
67 views

Solve $z^6+7z^3-8=0$

I want to find the solutions $z^6+7z^3-8=0$ but I don't know where to start because of the high degree of the equation. This is an exercise that involves complex numbers, so I have to transform the ...
2
votes
4answers
63 views

Show that no line with a y-int of 10 will ever be tangential to the curve with $y=3x^2+7x-2$

Show that no line with a y-int of 10 will ever be tangential to the curve with $y=3x^2+7x-2$. Having trouble in showing this. So far these are my process. Let line be $y=mx+10$ $mx+10 = 3x^2+7x-2$ ...
3
votes
3answers
104 views

Need help solving $x^4-3x^3-11x^2+3x+10=0$

Solve $x^4-3x^3-11x^2+3x+10=0$ I have tried to solve this equation using 'general formula from roots' from https://en.wikipedia.org/wiki/Quartic_function. $$ax^4+bx^3+cx^2+dx+e=0$$ $$x_{1,2}=-\frac ...
2
votes
1answer
25 views

What is the solution for $y(t)=e^{-\frac{t}{\tau y(t)}}$?

A simple quadratic flow model leads to the following apparently simple equation $$y(t)=e^{-\frac{t}{\tau y(t)}}$$ where the flow, $y$ is a function of time, $t$ and $\tau $ is a constant. But is ...
0
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1answer
27 views

If $x^4 + 3\cos(ax^2 + bx +c) = 2(x^2-2) $ has two solutions with $a,b,c \in (2,5)$, then find the maximum value of $\frac{ac}{b^2} $

The answer given is 1. i tried like this $3\cos(ax^2 + bx +c) = -x^4 +2x^2-4 = -(x^2 -1)^2 -3 $. The maximum value of $-x^4 +2x^2-4$ is $-3$ so $3\cos(ax^2 + bx +c) =-3$ and the two values of x are $...
0
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0answers
41 views

Testing if a Point is inside an oriented ellipsoid

So my goal is to do a search for the points that are inside an ellipsoid. for that i need to test if each point is inside. I need to be able to orient my ellipsoid, so i looked up how to write a ...
0
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2answers
49 views

Can't find my mistake

I'm trying to find the sum of the reciprocal numbers of squares of quadratic equation:$3x^2-14x+6=0$, I managed to find the answer by calculating the roots, and summing their reciprocal. However ...
0
votes
1answer
67 views

Equations $ax^2+btx+c=0, bx^2+ctx+a=0$ and $cx^2+atx+b=0$

Find different real numbers $a,b,c,t$ for which the following conditions: 1) the equation $ax^2+btx+c=0$ has real roots $x_1,x_2$; 2) the equation $bx^2+ctx+a=0$ has real roots $x_2,x_3$; ...
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2answers
22 views

Determine a, b & c in a (possible) Quadratic equation [closed]

given f(x)=x: 1 - f(5)=2 2 - f(3)=3 how to determine the coefficients a, b & c in the polynomial equation like: ...
0
votes
3answers
102 views

max of $e$ with $a+b+c+d+e=8$ and $a^2+b^2+c^2+d^2+e^2=16$ [closed]

Given that a,b,c,d,e are real number such that: $\begin{cases} a+b+c+d+e=8\\ a^2+b^2+c^2+d^2+e^2=16 \end{cases}$ determine the maximun value of $e$. I started like that : $\max(e)=\max(8-a-b-c-d)=\...
0
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1answer
28 views

Find set of values for $k$ with two distinct roots?

Find the set of values for $k$ for which the line $y=2x-k$ meets the curve $y=x^2+kx-2$ at two distinct points. I have started my equation like this: $$x^2+kx-2=2x-k$$ but I need to put it in the ...
0
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0answers
13 views

Indefinite Boolean Quadratic Programming: number of minima

The Boolean Quadratic Programming problem is defined as: $\min_{x} f(x) = x^TQx + c^Tx$ s.t. $ x \in \{0,1\}^n$ It is a well-studied NP-Hard problem with many approximation algorithms proposed. I ...
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2answers
31 views

Figuring $x$ and $y$ from two linear equations

I have a mini exam in a month to study for and I'm looking at systems of equations at the moment. I have this question to look at right now: Find $x$ and $y:$ $x-5y+4=1$ $\dfrac{x+1}{2}=y^2$ Now ...
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1answer
36 views

How to expand a equation into a quadratic equation? [closed]

I need a explanation and steps how I can expand this equation: $$(x−p)^2+(mx+c−q)^2=r$$ into this one: $$(m^2+1)x^2+2(mc−mq−p)x+(q^2−r+p^2−2cq+c^2)=0$$ Thank you for your time.
2
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2answers
30 views

Find the set of values of x for which $\frac{x+1}{2x-3}<\frac{1}{x-3}$

Here's what I've done: $\frac{x+1}{2x-3}<\frac{1}{x-3}$ $x+1<\frac{2x-3}{x-3}$ $(x+1)(x-3)<2x-3$ $x^2-2x-3<2x-3$ $x^2-4x<0$ $x(x-4)<0$ $0<x<4$ However this clearly ...
3
votes
1answer
46 views

Solution to a simple system of quadratic equations

I am hoping to find a closed-form solution to the following system of $n$ quadratic equations: $$ x_j^2 = \sum_{i=1}^n B_{ij}x_i $$ for $j\in\{1,\dots,n\}$, where $B_{ij}\geq 0$. There is a trivial ...
0
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0answers
20 views

Involves finding minimum length as x varies

Ok in a question where you have been given coordinates for a and b (-3,5),(x,3x+9) respectively and AB^2 is given to be 10X^2+30x+25 for this 5 is taken out as a factor so you're left with 5(2x^2+6x+5)...
1
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0answers
29 views

Am I finding this $x$-value correctly?

If the flight path of a cricket ball is given by: $$y = \frac{1}{3}x - \frac{1}{60}x^2$$ And a fielder standing originally at $(10, 0)$ catches the ball when it is $1.5$ units above the ground, to ...
2
votes
1answer
43 views

$f(x)$ is a quadratic polynomial with leading coefficient $1$, $|f(x)| \leq 8 \: \forall \: x \in [1,9]$ find $f(x)$

$f(x)$ is a polynomial of the form ($b,c$ are real numbers) $$f(x) = x^2+bx+c$$ such that $|f(x)| \leq 8 \: \forall \: x \in [1,9]$. Find all $f(x)$ satisfying the given condition. I found ...
0
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1answer
52 views

How do you solve a system of equations with e^x

How do you solve a system of equations with e^x. For example
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2answers
35 views

$100 + [110/(1+r)] = [1/ (1+r)] + [(232 /(1+r)^2 ]$

Need to learn how to solve this: $100 + \frac{110}{1 + r} = \frac{1}{1 + r} + \frac{232}{(1 + r)^{2}}$. Checked this site got to the 3rd line and am completely lost. Can someone help me solve for r ...
0
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3answers
54 views

finding rational roots

Consider the integral expression in $x$ $$P=x^3+x^2+ax+1,$$ where $a$ is a rational number. At $a= ?$ the value of $P$ is a rational number for any $x$ which satisfies the equation $x^2+2x−2=0$, and ...
0
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1answer
18 views

Total number of integral solutions for the given second degree equation!

First, the problem statement : "Consider the equation $x^{2}+y^{2}-3z^{2}-3t^{2}=0$. The total number of integral solutions of this equation in the range of the first 10000 numbers, i.e., $1\leq x,y,z,...
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1answer
63 views

Range of the expression $\frac{9 \cdot 3^{2x}+6\cdot 3^{x}+4}{9 \cdot 3^{2x}-6 \cdot3^{x}+4}$ [closed]

Given that, for all $x \in \mathbb{R}$ the expression $\frac{x^2-2x+4}{x^2+2x+4}$ lies between $1/3$ and $3$ the values between which the expression $\frac{9 \cdot 3^{2x}+6\cdot 3^{x}+4}{9 \cdot 3^{...
4
votes
1answer
194 views

Can we use matrix to solve this inequality?

Let $$f(x)=\begin{cases} 1&0\le x\le 1\\ 0&\rm{others} \end{cases}$$ Let $x_{i},a_{i}(i=1,2,\cdots,n)$ be positive real numbers, show that: $$\sum_{i,j=1}^{n}a_{i}a_{j}f\left(\dfrac{|x_{i}-...
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votes
2answers
48 views

Does a square root come out plus/minus even if there is a negative sign outside?

For example: $-\sqrt{100x^{20}y^{10}}$. Would that give $\pm10x^{10}y^5$ or just $-10x^{10}y^5$?
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1answer
46 views

Fit a Quadratic Curve to Data

I have some data and I want to fit a quadratic curve for my data But I don't know that how to it do? My data : $x,y = 100,45;$ $x_1,y_1= 101, 50$; $x_2,y_3=99,35$; $\ldots$ For instance this ...
0
votes
0answers
20 views

How to use the complete the square method for a given function.

I am given a function $u(y_1, y_2, y_3)= (y_1)^2+(y_2)^2+(y_3)^2-2y_1+2$ for $y$ on the boundary of the ball $B(x,2)$ and it ends up that this function is equal to $u(y) = |y-x|^2 +1$, $y$ and $x$ are ...
0
votes
1answer
37 views

If $ a(12a + 5b +2c) > 0 $ , then prove that the real roots of equation $ ax^2+ bx +c =0$ are less than 2

I know that to prove both roots less than 2 ,i have to prove $ 4a+ 2b +c > 0$ and $ -b/2a < 2 $ . Here i have no idea how to proceed.
0
votes
1answer
40 views

Are there any tricks for simultaneous equations I should be aware of?

I'm at the end of a difficult logarithms question and have ascertained the linear equations I need in order to establish x and y as the questions asks of me. The equations are: $x - 5y + 4 = 1$ $\...
3
votes
1answer
52 views

Roots of the equation $x^2+1=0$ in $\Bbb Z/p^{n}\Bbb Z$

Let $p$ be an odd prime number and $n$ be a positive integer. I want to consider roots of the equation $x^{2}+1=0$ in the ring $\Bbb Z/p^{n}\Bbb Z$. Suppose $n=1$. Find a condition on $p$ such ...
0
votes
0answers
38 views

How can you solve a polynomial with a power of 1.99?

$y_2 = -.0251256t^{1.99} + (v_{y})t + y_1$ How would I solve for $t$? Ideally I would approximate as a quadratic, but the error becomes too high.
4
votes
10answers
156 views

Factor $6x^2​ −7x−5=0$

I'm trying to factor $$6x^2​ −7x−5=0$$ but I have no clue about how to do it. I would be able to factor this: $$x^2-14x+40=0$$ $$a+b=-14$$ $$ab=40$$ But $6x^2​ −7x−5=0$ looks like it's not ...
3
votes
1answer
36 views

When are we able to find a quadratic with roots that are a function of another quadratic?

Motivation: Given the roots of the quadratic $2x^2+6x+7=0$ find a quadratic with roots $\alpha^2-1$ and $\beta^2-1$ I was able to solve this problem in two ways: Method 1: Sum of the roots $\alpha+...