Questions about quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$.

3,653 questions
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Determining whether a quadratic has a maximum or minimum

So I've learnt that quadratics equations with a positive coefficient on the squared term have a minimum and a maximum if the coefficient is negative. But if we rearrange the quadratic and change the ...
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Is it true that $(a^2-ab+b^2)(c^2-cd+d^2)=h^2-hk+k^2$ for some coprime $h$ and $k$?

Let us consider two numbers of the form $a^2 - ab + b^2$ and $c^2 - cd + d^2$ which are not both divisible by $3$ and such that $(a, b) = 1$ and $(c,d) = 1$. Running some computations it seems that ...
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Simplification of identity with square roots [closed]

$\dfrac{\sqrt{x + 1}}{2x + 1} + \dfrac{\sqrt{2x + 1}}{x + 1} = 1 \tag 1$ How can I find the value of $x$ in this question?
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Change the vertex of a parabola while ensuring it still passes through a particular point

I have a parabola defined by the quadratic equation $y = -(x + 0)(x - endPoint)$, which also passes through a particular point $(a, b)$. I would like to know how to alter the equation so that I can ...
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How would I go about finding the steady states I know I need to set $\frac{dx}{dt}=0$ but then I'm struggling with the next step.
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If $\alpha$, $\beta$ are the roots of the equation $ax^2 + 3x + 2 = 0,\; a<0$

Show that if $\alpha$, $\beta$ are the roots of the equation $ax^2 + 3x + 2 = 0,\; a<0$, then $$\dfrac{(\alpha^2)}{(\beta)}+\dfrac{(\beta^2)}{(\alpha)}> 0$$ I could only figure out two things ...
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Solving simple quadratic - Wolfram Alpha confusion?

I have the following quadratic $$(2\sqrt 2 - 2)x^2 + \sqrt8 x + (1+\sqrt 2)=0$$ Now the discriminant of this is $0$, so it has one real repeated root. A plot on Desmos confirms this. However, ...
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If the equation $\sin^2x-a\sin x+b=0$ has only one solution in $(0,\pi)$, then what is the range of $b$?

If the equation $\sin^2(x)-a\sin(x)+b=0$ has only one solution in $(0,\pi)$, then what is the range of $b$? What I try: $$\displaystyle \sin x=\frac{a\pm \sqrt{a^2-4b}}{2}\in\bigg(0,1\bigg]$$ for ...
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How to find quadratic regressions by hand

Currently I am working on an assignment for which I have to calculate the quadratic regression and linear regression (I know how to do this one) of some data points by hand. Nonetheless, I do not know ...
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Solving steps for equation with exponent and addition (x^2 + x) = 2y

I have this equation: ($x^2$ + x) = 2y Which I know solves to: x = (-1 + sqrt(1 + 8y)) / 2 x = (-1 - sqrt(1 + 8y)) / 2 However, I have no idea about the steps to reach the solved equation, any ...
If If $x^2+ax+b+1=0$ has roots which are positive integers, then $(a^2+b^2)$ can be which of the given choices?
If $x^2+ax+b+1=0$ has roots which are positive integers, then $(a^2+b^2)$ can be (A) 50 (B) 37 (C) 19 (D) 61 My approach: I first took roots $\alpha$, $\beta$ and then applied sum and ...
Who was the first person to prove that only primes of the form $4k+1$ can evenly divide odd integers of the form $n^2+1$?
I am writing a paper and I want to cite the person(s) who proved that only primes of the form $4k+1$ can evenly divide odd integers of the form $n^2+1$. Edit: added "odd" For example, if $n=8$, then ...