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 $y=a(x+b)(x+c)$.

3,674 questions
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### What is the Hessian w.r.t. to matrix X of this quadratic function?

I am stuck in finding the Hessian w.r.t. to matrix $X \in R^{m \times n}$ in the following : $$\frac{1}{2} ||AXB-C||_F^2$$ where $A \in R^{l \times m}$ and $B \in R^{n \times o}$ I got the first ...
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### Function with 2 unknowns and one needs to be solved [closed]

I have a homework question stating: Find the set of values of $k$ for which $f(x)=3x^2-5x-k>1$ for all $x\in\mathbb R$. This question has really confused me because I looked at the answer and ...
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### How to deal with $x^T(A^TA+B^TB)x$?

I want to minimize $$\|Ax-b\|_2^2 + \|Cx-d\|_2^2 + \|x\|_1$$ I know we can introduce an auxiliary variable and use ADMM to separate the $\ell_2$ and $\ell_1$ norms. But is there a more convenient ...
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### Generating function with multiple variables with perfect square

Consider trinomial $x^2 + bx + c = 0$. When $b = 205$ and $c = -206$, then $x = 1$ and the function evaluates to a perfect square. My question is, how can I choose $b$ and $c$ and generate functions ...
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### $3x^2+5xy-2y^2-3x+8y+\lambda$ How to express as two factors

$3x^2+5xy-2y^2-3x+8y+\lambda$ find appropriate value for $\lambda$ such that expression can be expressed as two linear factors. My Try First I thought of separately writing x terms as complete ...
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### How to properly write a solutions of quadratic equation on one line?

Using this formula $$x_{1,2}={-b\pm\sqrt{b^2-4ac} \over 2a} = \space \cdots$$ what should be the correct format of the solution if I want to put both roots (x) on one line? Thanks in advance
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### Sum of all the possible real solutions of $(x^2+4x+5)^{(x^2+4x+5)^{(x^2+4x+5)}}=2018$

Sum of all the possible real solutions of $(x^2+4x+5)^{(x^2+4x+5)^{(x^2+4x+5)}}=2018$ My try I knew that the answer is $-4$ (according to wolfram alpha) but i tried myself solving it by hand. I ...
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### Condition for both roots be infinity [closed]

For what value/s of constant 'p' for which the given quadratic have both roots as infinity. $(2p^3-13p^2+27p-18)x^2 + (2p^2-9p+9)x +2p^2-7p+6=0$ Options are :- $1) 3/2 2) 2 3) 3 4) /phi$ Since both ...
I am young and try to solve this quadratic equation: $x^2 + 6x +5 = 0$ How can I solve it?