# 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 $m \in \mathbb{Z}$ for which $x_1$ and $x_2$ are integers

$$(m+1)x^2 - (2m+1)x - 2m = 0$$ $$m \in \mathbb{R}-\{-1\}$$ Find $m \in \mathbb{Z}$ for which $x_1$ and $x_2$ (the solutions of equation, the roots) are integers ($x_1,x_2 \in \mathbb{Z}$) ...
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### On the solution of the equation $z+ \alpha \left| z-1\right| + 2i = 0$

Question:- Find the range of real number $\alpha$ for which the equation $z+ \alpha \left| z-1\right| + 2i = 0$; $z=x+iy$ has a solution. Also find the solution. Attempt at a solution:- On ...
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### Can a quadratic have non-zero coefficients when the roots are $k$ and $-k$?

Can a polynomial like this: $$ax^2+bx+c$$ have two opposite roots, without either $b$ or $a$ being equal to zero?
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We know convex combination of concave/convex functions are concave/convex. While convex combination of two quasi-convex/quasi-concave functions necessarily quasi-convex/quasi-concave. Common ...
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### Find m so that the equation has integer solutions

We are given the following equation: $(m+1)x^2-(2m+1)x-2m=0$, where $m\neq-1$. We have to find all integers $m$ so that the equation above has integer solutions. I know that $m=0$ and $m=-2$ satisfy ...
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### Solving Equation With Logarithm Argument Being a Variable

$$8n^2 = 64n\log_{2}n$$ Been a while since I have used logarithms. I am actually comparing the running time of two algorithms and can obviously solve graphically but for the life of me can't remember ...
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### Prove $(1+c/a+|b/a|)<0$ from $(a+b+c)(a-b+c)>0$

While doing a certain sum I got stuck at a step.I am getting $(a+b+c)(a-b+c)>0$.I need to prove $(1+c/a+|b/a|)<0$.Is it possible?How? a is not 0. The original question : If $ax^2+bx+c=0$ has ...
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### show the quadratic function $W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j$ is quasi-concave

I have a quadratic function $W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j$, with $x_i$ nonnegative and $A \in[0,1)$. And w.l.o.g. we can normalize $x_i's$ to between 0 and 1. In ...
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### Prove that $b^2/q^2=ac/pr$?

Let $\alpha_1,\alpha_2$ and $\beta_1,\beta_2$ be the roots of the equation $ax^2+bx+c=0$ and $px^2+qx+r=0$ respectively.If the system of equations $\alpha_1y+\alpha_2 z=0$ and $\beta_1 y+\beta_2 z=0$ ...
Let $a,b,c,d$ be real numbers in G.P.If $u,v,w$ satisfy the system of equations $u+2v+3w=6$, $4u+5v+6w=12$, $6u+9v=4$ then show that the roots of the equation \left(\frac{1}{u}+\frac{1}{v} + \...
### $'a'$ for which roots of one equation lie between roots of other equation.
If the range of values of $'a'$ for which the roots of the equation $x^2-2x-a^2+1=0$ lie between the roots of the equation $x^2-2(a+1)x+a(a-1)=0$ is $(p,q)$, find the value of $(q+\frac{1}{p^2})$ ...