# 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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### Find all rational points where $x^2 - y^2 = 1$ (need help simplifying quadratic formula) [duplicate]

The original problem is to find all rational points where $x^2 - y^2 = 1$ I know how to go about the problem, but whenever I get to the point of simplifying my equation, I keep having problems. This ...
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### Quadratic function that produces natural number from natural number inputs

I am currently trying to find a way to generate different (preferably quadratic) function as part of a encryption algorithm such that : ...
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### Roots of $f(x)+g(x)$

Question : Let $p,q,r,s \in \mathbb R$ such that $pr=2(q+s)$. Show that either $f(x)=x^2+px+q=0$ or $g(x)=x^2+rx+s=0$ has real roots . My method : To the contrary suppose that both $f(x)$ and $g(x)$...
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### Why not always use the quadratic equation

The is a very simple question, but I have just started studying quadratics. I understand how to factor them using different methods and also understand solving a quadratic using the formula, but my ...
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### Find integers $x$ and $y$ such that $\frac{27^{x+y}}{9^{xy}}=27$ and $\frac{4^{2xy}}{8^{x+y}}=512$ .

Find all the integers $x$ and $y$ such that : $$\frac{27^{x+y}}{9^{xy}}=27$$ and :$$\frac{4^{2xy}}{8^{x+y}}=512$$ I'm in Algebra two and I feel like there are certain types of math I haven't ...
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### How can you find $m$ in $mx^2+(m-3)x+1=0$ so that there is only one solution

How can you find $m$ in $$mx^2+(m-3)x+1=0$$ so that there is only one solution. I tried to solve it by quadratic equation but I end up with two solutions. So I want it know that is there a way so ...
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### Can this nonlinear simultaneous equation be solved?

Problem: $\{A,B,E,R,S\}\in R^{n \times n}$ are square matrices, $\{\mathbf{x},\mathbf{y}\}\in R^{n}$ are vectors. Particularly, $\{ A,B \}$ are symmetric matrices, and $E$ is an identical matrix. We ...
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### Solve the following number theory problem with 2 variables [closed]

Let there be $$a,b∈ \Bbb Z$$ Demonstrate that there exist no solutions for the following equation $$a^2-3b^2=-1$$
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### How do the roots of “$x^2 + bx + c$” change as $b$ is kept constant and $c$ is changed? [closed]

Consider the function $x^2 + bx + c$ How do the (real or complex) roots of the equation change if $b$ is held constant and $c$ is changed? I.e. Which patterns are evident? What would it look like if ...
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### Show that $p(x)=rq(x)$ for some rational number $r$.

Let $p(x)$ and $q(x)$ be two quadratic polynomials with integer coefficients. Suppose they have a non rational root in common. Show that $p(x)=rq(x)$ for some rational number $r$.
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### Calculate the quadratic residues in Z∗17.

Hello I am wondering if any one can help me I am trying to figure out how these below answers where came to too. Calculate the quadratic residues in Z∗17. Solution: This can be done by direct ...
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### Looking for a function $g(x)$ such that $g(2x+2) = g(x) + 2x+2$

So recently I got bored in maths class (I'm in tenth grade) and made up a little equation that looked something like this: $$g(f(x)) = g(x) + f(x)$$ My original goal was to find different $g(x)$ to ...
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### No. of parabolas possible for the given equation.

Given, $f(x)=-x^2+qx+r$. $(q,r) \epsilon R$. $q,r$ are variables. A quadratic equation $f(x)=0$ has a maximum value $m$ ($m$ is a constant) and a root $x=a$. Does $f(x)$ correspond to a unique ...
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### Easy word problem but am I working it too hard?

A friend of mine gave me this problem: A man who walks at a constant speed goes to his barn 30 miles away with a 2 mph wind pushing against him. After arriving at the barn he remembers he ...
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May I know how do I form a quadratic number pattern equation? I cant seem to form one on my own. 1500, 1519,1536, 1551,1564.
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### Did I fully explain this optimization and quadratics problem?

I'm not really sure how to explain the last part; how does solving for $x$ by replacing $y$ show that $x^2+y^2$ is greater than or equal to $9?$ Like, I get why, but I don't know how to express it. ...
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### How to find a quadratic equation given three points, two on the x-axis?

Find the quadratic equation for a parabola that passes through $$(1,0) (5,0) (0,10)$$ To do this I turned it into $$x = 1$$ $$x = 5$$ and then into $$(x-1)(x-5)$$ after you multiply everything ...
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### How to solve the following quadratic word problem given a quadratic equation?

The height of a ball(h), in feet, after s seconds is modeled by the equation $$h=-16t^2+40t-6$$ How many seconds does it take for the ball t reach its maximum height? First thing i did was turn the ...
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### Find quadratic equation given two points and y-intersection

Let the domain be $x \in [0,h]$. We have three points, $(0,1)$, $(h/2,0)$ and $(h,0)$. How do I find the quadratic equation? My attempt: I know that the roots are located at $x=h/2$ and $x=h$. Thus ...
I have a question about why the unknown becomes absolute when taking the square root in an inequality. For example: Find the value(s) of $k$ for which the equation $2x^2-kx+3=0$ will have two ...
### If $x-y = 5y^2 - 4x^2$, prove that $x-y$ is perfect square
Firstly, merry christmas! I've got stuck at a problem. If x, y are nonzero natural numbers with $x>y$ such that $$x-y = 5y^2 - 4x^2,$$ prove that $x - y$ is perfect square. What I've ...