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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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
votes
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 ...
-2
votes
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 ...
-1
votes
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
votes
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 ...
4
votes
1answer
48 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
votes
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
vote
0answers
30 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
votes
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
-1
votes
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
votes
3answers
55 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
votes
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,...
-6
votes
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
197 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}-...
-1
votes
2answers
49 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 $-10x^{10}y^5$?
1
vote
1answer
47 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
53 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
158 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
2answers
40 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+...
2
votes
2answers
85 views

If $\alpha_1,\alpha_2,\ldots,\alpha_n$ be the roots of the equation $x^n+1$

then $(1-\alpha_1)(1-\alpha_2)\ldots(1-\alpha_n)$ equals to ? I think here we need the info of whether $n$ is even or odd else how will we say whether by vieta's formula what is the value of $1+(-1)^n$...
0
votes
2answers
15 views

How to find the solutions for the quadratic equation for conic sections $\epsilon \in (0,1)$

Going from this definition of the conic section: $\epsilon |Pl| =|PB|$, you get the following equation for the intersection with the $x$-axis: $y^2 = (\epsilon ^2-1)x^2+(B-\epsilon ^2L)2x+\epsilon ^2L^...
1
vote
1answer
48 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?
2
votes
0answers
43 views

question on quadratic expansion [closed]

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 $\dfrac{a+b}...
3
votes
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 + 2*\sqrt{x+938^2}*\sqrt{...
1
vote
1answer
23 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.
0
votes
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 $ ...
1
vote
1answer
29 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= \frac{x+a}{...
0
votes
2answers
25 views

Value of qudaratic equation

In my exams, I was asked to calculate value of Quadratic Equation from given value of a, b, c...
0
votes
0answers
16 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 ...
0
votes
1answer
31 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 - ...
0
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0answers
9 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 ...
0
votes
1answer
29 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 ...
1
vote
2answers
26 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 ...
0
votes
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 Gal$(F(x)/...
3
votes
1answer
73 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): Then,...
0
votes
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
197 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
96 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
230 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
votes
3answers
35 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
35 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 ...