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
40 views

Factoring out an equation

I found this equation in a book: $$ m_0 v_0^2 + m_1 v_1^2 = m_0 v_{0_{Final}}^2 + m_1 v_{1_{Final}}^2.$$ It says that Notice that you have a different equation with the same two unknown ...
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1answer
17 views

How-To Quadratic Funcions and Graphs

I have the following problem: For $f(x) = −x^2 + 4x − 8$ the value of $-b\over{2a}$ is $2$. Find the $y$-coordinate of the vertex of the graph of this function. My book is severely lacking in ...
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1answer
40 views

Solving a quadratic expression

I have an equation of the form: $$x'Ax - B$$ where $A$ is a positive definite matrix. I want to solve this equation for $x$. Could anyone provide me with some suggestions on how to solve this kind of ...
0
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1answer
21 views

Minimizing a quadratic equation, with constraint.

I have a problem, I want to minimize this: $$ \min_{w} {w^\dagger H_1 w}\\ s.t. {w^\dagger H_2 w} = \mu^2 \\ ||w||^2 = 1 $$ with $\mu$ being real positive number, and $H_1, H_2$ are matrices with same ...
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1answer
18 views

Misstep With Discriminant and equations writable as quadratic form

I missed a step in my equation and would like to know what i'm doing wrong. I have the following equation: ${x^{4} - {\color{red}15} x^{2} + {\color{red}54} = 0}$ Now, we let ${y = x^2}$ We can ...
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2answers
41 views

Polynomial Equations Root Finding

If one of the roots of the Equation : $$2000x^6 + 100x^5 + 10x^3 + x - 2 = 0$$ ; Is of the form $$(m + √n)/r$$. Where m is a non zero integer and n and r are relatively coprime. We have to find ...
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1answer
22 views

quadratic equation problem - proving a statement

I was given that $ax^2+2bx+c=0$ Using $y=x+\frac{1}{x}$ I need to prove that $acy^2+2b(c+a)y+(a-c)^2+4b^2=0$ Tried to make the pattern $x+\frac{1}{x}$ and to substitute $y$, but couldn't prove it.
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0answers
68 views

New way to solve Quadratic Equations?

A couple months ago, I was challenged to discover a new way to solve quadratic equations with one rule: I must think of it myself. At first I was hopeless, trying various sorts of random things, but I ...
2
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0answers
55 views

Equation over free group

Let us consider free group $F(a,b)$ of rank two. I need to find a solution (or to prove that there is no one) over this group of the following equation: $$x^2[x^{2k},y]=a^2b^2,$$ where ...
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0answers
28 views

Perfect squares in $S_i = i^2 + b \cdot i + c$

I need to find perfect squares ($S_i = x^2$) in sequence: $$S_i = i^2 + b \cdot i + c$$ where all numbers are natural: $$i,b,c,x \in \Bbb{N}$$ Now I use a straightforward approach, i just iterate ...
1
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1answer
32 views

Roots of quadratic equation by completing the square or other method?

I'm trying to find solution(s) to the following equation: $x^2 - 5x + 3 = 0$ It seems like it can't be factored normally so I tried solving by completing the square: $x^2-5x=-3$ $x^2-5x+6.25=-0.5$ ...
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3answers
55 views

form a quadratic equation whose roots are $2α+β$ and $α+2β$

$g(x)=x^2+kx+2k-3$, where $k$ is a constant. Given that $g(x)=0$ has roots $α$ and $β$, form a quadratic equation whose roots are $2α+β$ and $α+2β$ Answer- $x^2+3kx+(2k^2+2k-3)=0$
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5answers
57 views

Given that the roots of the equation $9x^2+bx+4=0$ are $4a$ and $a$ and that $b>0$, find the value of $b$.

Given that the roots of the equation $9x^2+bx+4=0$ are $4a$ and $a$ and that $b>0$, find the value of $b$.
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3answers
47 views

What is the solution of this quadratic function

$$iz^2 − 2 \sqrt{2}z − 2 \sqrt{3} = 0,\qquad z \in \mathbb{C}$$ I'm new to this site, i'm really thankful for your help
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1answer
24 views

When the equation of a conic becomes that of a pair of straight lines

This is a question I found in a book. Let $0<p<q$ and $a\neq0$ such that the equation $$px^2+4\lambda xy+qy^2+4a\left(x+y+1\right)=0$$ represents a pair of straight lines, then $a$ can lie ...
1
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2answers
112 views

Quadratic Function Help

Not sure how to solve this one: State the value $k$ such that the function $f(x) = -2 (x-3)^2 + k$ has $1$ root. So I already know that this has to be a sideways parabola, but don't know how ...
6
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2answers
45 views

Show $p$ prime s.t. $p \not\equiv 1 \mod 3$ is represented by the binary quadratic equation.

I am working on the following question: Let $p>3$ be a prime such that $p \not\equiv 1 \mod 3$. Show that $p$ is not represented by the binary quadratic equation $f(x, y) = x^2 + xy + y^2$. I ...
0
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1answer
34 views

Solving a Binomial Sextic

Say I have a sextic equation, but I'm able to get it into the form: $$ax^6 + dx^3 + g = 0$$ I know that I can do a simple substitution like $y = x^3$ to get an equation that I can solve with the ...
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4answers
99 views

Finding the values of $5a+b$ if $ax^2+bx+10$ does not have $2$ distinct roots

If $ax^2+bx+10=0$ does not have two distinct real roots where( $a$ and $b$ are real) then the possible values of $5a+b$ from the given 4 options(as obviously there can be infinite possible values..but ...
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2answers
33 views

Finding common interest rate

My question is; How can I approach a common interest rate? (See example below for an explanation) Consider the example in the table below ...
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2answers
54 views

How to prove that the roots of $(m-2)x^2-(3m-2)x+2m=0$ are real

I need help proving that the roots of the equation: $(m-2)x^2-(3m-2)x+2m=0$ are real. Could you also give me a step by step runthrough of how to do this equation? (I have a few others like this one to ...
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0answers
11 views

Is this the correct way to interpret results from the principal minor test?

Am I correct in thinking that, given all the principal minors $A_{i}$ of the matrix associated with a quadratic form $Q(x,y)$, the Principal Minor Test states that $Q(x,y)$ is Positive Definite if ...
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0answers
19 views

How Do I Obtain X from this equation?

I tried different ways, but I don´t succeed: $y=\frac{4x-1}{8x^4}$ In one of them I reach this point: $4x(2x^3-1)=-\frac{1}{y}$
2
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1answer
47 views

Prove a binary quadratic equation has specific number of solutions

How do I show that the binary quadratic equation $f(x, y) = x^2 + xy + y^2 = 1$ has exactly $6$ solutions? The discriminant is $-3$, so I cannot use Pell's Equation ($x^2 - dy^2 = p$, where $d>0$ ...
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4answers
45 views

Basic quadratic calculus

Given $$ AX^2+2X-1=0 \,\ \text{where} \,\ A>0$$ What value of A would make the absolute value of both roots bigger than $1$? I found the roots using quadratic formula and also found that A has to ...
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0answers
52 views

Reference book for quadratic equations with hard exercises?

I'm teaching advanced level mathematics. And I need good question banks on quadratic equations . More likely good question bank on advanced level maths.
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0answers
28 views

Problem on Bivariate normal distribution

Let $X_1$ and $X_2$ have a bivariate normal distribution with parameters $\mu_1 = \mu_2 = 0$ and $\sigma_1 = \sigma_2 = 1$ and $\rho = 1/2$ Find the probability that all the roots of $X_1x^2+ 2X_2x + ...
1
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0answers
31 views

How to minimise objective function with H1 constraint?

I have a cost function that I try to minimise: $$ \Pi \sim ||y - Ax||^2 + ||\nabla x || ^2 $$ The gradient is there to constraint that my solution at minimum $x_{min}$ is continuous in 1st ...
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2answers
33 views

Correct notation for inequality equation

I'm currently taking a Polynomial functions -course and learning about quadratic inequality equations. Let's consider this equation: $-x^{2}+7x<10$ The correct answer to this according to our ...
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2answers
83 views

Need help with a certain integral

Sorry, this is my first post ever and formatting is bad. I appreciate all assistance Given the equation y= $ 3.8x^2-4.4x+1444 $ Find the definite arc length integral between 0 and 1200. I am not ...
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0answers
44 views

Find root of $x^2 + bx + c$ with an approximation for $b$

I have a monic quadratic equation $x^2 + bx + c$, where $b, c \in \mathbb{Z}$ for which I need to compute the roots. $x$ is also an integer. However, I don't have the exact value of $b$, only an ...
1
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2answers
24 views

Showing solutions for this recurrence relation

I am having trouble starting this question. Given that the sequence ($x_n$) satisfies the recurrence relation: $x_n$$_+$$_1$ = a$x_n$ + b$x_n$$_−$$_1$, where $n= 1, 2, ...$ and $a$ and $b$ are given ...
0
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0answers
21 views

Solving non-commutative “quadratic” equation with inhomogenously typed coefficients

Is there a general method to solve for $z\in\mathbb{R}^d$ in the non-commutative $z^\intercal\alpha z + \beta z + \gamma = 0$ where $\alpha\in\mathcal{M}_d(\mathbb{R})$ (real $d\times d$-matrix), ...
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1answer
64 views

How to solve a system of quadratic equation as follows?

I have a system of quadratic equations: \begin{align*} v_2x_1^2 + v_2x_1 - v_1x_2^2 - v_1x_2 & = 0\\ v_3x_1^2 + v_3x_1 - v_1x_3^2 - v_1x_3 & = 0\\ v_3x_2^2 + v_3x_2 - v_2x_3^2 - v_2x_3 & = ...
1
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0answers
42 views

Does an equation of this type have complex solutions?

How can I show, that every quadratic equation of the type $z^2+az+b=0$ with complex coefficients $a,b$ has a solution in $\mathbb{C}$? And how can I get these solutions? At first, I tried applying ...
0
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2answers
46 views

solution of quadratic equation n unknown

My question is as follows: Let the equation $V^{\top}MV=F$. Such as $V^{\top}=(x_1,x_2,...,x_n)$ a line vector of n unknown coefficients, M a known diagonal matrix (of size n) and F a real number ...
0
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1answer
16 views

Conditions present after solving for x

The question asked me to solve $\dfrac {x^2+2x-8}{x^2-x-2}=3$ The answer is $x=0.5$, which I worked out, but in the answers it says that I must state the condition that $x≠2$ to get full marks. Why ...
5
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3answers
126 views

Solving $10x^4-13x^2+4=0$

I just came across a question in my paper that asks me to solve for $x$ in $10x^4-13x^2+4=0$ I've only learned how to factorize quadratics and the quadratic formula, but I'm not sure how to factorize ...
0
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3answers
16 views

Factorising quadratic equation with an unknown given 1 root

In a algebra quadratic question, it says: The equation $3x^2+4x-k=0$ has two distinct real roots. If 2 is a root of this equation, find the value of $k$ and the second root. The first line of ...
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0answers
21 views

Principle axes rotation.

I am trying to rotate a quadratic function to its principle axes. I have transformed the function to a symmetric 3x3 matrix I am struggling to know what to do next I thought to rotate the matrix but ...
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1answer
28 views

Rewriting an algebraic equation with square roots

In as part of solving a question, the equation $a-3\sqrt a-4=0$ is written into $a^2-3a-4=0$ How is this done? Do you square everything in the equation? But in this case why are only the $a$ ...
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0answers
11 views

Keeping a parabola's roots after vertical shift

Suppose f(x) = -x(x - a) + b, where a > 0 and b >= 0. When ...
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4answers
90 views

How to Solve $\frac{(x^2+1)}{x} + \frac{x}{(x^2+1)} = 2.9$?

My cousin has this math homework problem she is stuck on and I have also failed at it. Maybe you guys can help? $$\frac{x^{2}+1}{x} + \frac{x}{x^{2}+1} = 2.9$$ The answer is $0.5$ or $2$, but we're ...
0
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0answers
18 views

polynomial roots and density

Let $f(x)=x^2+2x$ be a quadratic polynomial and $\mathcal{R}_f$ be its range. Denote $\mathcal{A}_{i+1}$ be the set of real numbers that lie in $\mathcal{R}_f$ and come from finding roots of ...
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0answers
37 views

Econometrics: quadratic specification, turning point and time-series

Given the following quadratic specification : ln(yt) = c + beta1*ln(xt) + beta2*ln(xt)^2 where t: represents time Ln: natural logarithmic c: constant yt: dependant variable at time t xt: ...
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0answers
13 views

Manipulation of this equation into a Gaussian form

I have been going through a paper (http://www.jting.net/pubs/2007/ting-ICRA2007.pdf) and trying to work out the maths. Ultimately, I came to the following expression $$ \bigg[\frac{1}{2 ...
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2answers
43 views

Is is possible for a quadratic equation with only one irrational root to have integral coefficients?

Given a quadratic equation with one and only one root (for example $6-\sqrt{2}$ ). Does there exist integers $a,b$ and $c$ where $ax^2 + bx + c = 0$ for that root?
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1answer
45 views

Quadratic equation involving right-angled triangle

I have right angled triangle I've been attempting to prove a quadratic equation with for a while. It has a hypotenuse of $2x + 1 cm$, a base of $x + 5 cm$, and height of $x - 2 cm$. I calculated its ...
0
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1answer
32 views

How to solve this equation for y? $y^2+2yx-3x^2=0$

How to solve this equation for y? $$y^2+2yx-3x^2=0$$ I know how to factor the unknown y with the coefficients of the expression and it would be $(y+3)(y-1)$ but I don't remember how to work out x. ...
0
votes
1answer
46 views

Quadratic function.

Let $\quad f(x)=ax^2+bx+c \quad $ be a quadratic function, $x \in \left (0,d \right )$ a domain of $f(x)$. If I need $f(x)$ to be positive, that would be achieved for $ x \in \left (-\infty,x_{-} ...