# Tagged Questions

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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### 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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### 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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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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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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 ...
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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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: $4x(2x^3-1)=-\frac{1}{y}$
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### 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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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 ...