Abbas Damerchi@abbasdamerchiOct 30THE FIN / SYN PARADOXyoutube.comBig Different Between SYN & FIN Scans | -sF Flag #34With Using -sF Flag You Be Able To Send FIN Proofs.In This Episod...121
Abbas Damerchi@abbasdamerchiOct 27Hello Grok, Teach me: Standard Forms factor in mathematics like: x^2 + 2 bx + cAnswer by GrokFactoring quadratic expressions of the form \(x^2 + 2bx + c\) A quadratic expression is typically written in the general form \[ax^2 + bx + c\] (with \(a \neq 0\)). Your example is the special case \(a = 1\): \[x^2 + 2bx + c.\] Perfect-square (completed-square) form The most important special case is when the constant term equals \(b^2\): \[ x^2 + 2bx + b^2 = (x + b)^2 \] (and likewise \(x^2 - 2bx + b^2 = (x - b)^2\)). How to see it Expand the right-hand side: \[ (x + b)^2 = x^2 + 2\cdot x\cdot b + b^2 = x^2 + 2bx + b^2. \] So whenever the middle coefficient is exactly twice the square root of the constant term (and the signs match), the expression is already a perfect square. Example \[ x^2 + 6x + 9 = (x + 3)^2 \] because \(2b = 6\) \(\Rightarrow\) \(b = 3\) and \(b^2 = 9\). Completing the square (when it is not already a perfect square) If \(c \neq b^2\), rewrite the expression by adding and subtracting \(b^2\): \[ \begin{align*} x^2 + 2bx + c &= x^2 + 2bx + b^2 - b^2 + c \\ &= (x + b)^2 + (c - b^2). \end{align*} \] This is the completed-square form. It is extremely useful for: solving equations, graphing parabolas (vertex form), integration, etc. Example \[ x^2 + 6x + 5 = (x + 3)^2 - 9 + 5 = (x + 3)^2 - 4. \] Factoring over the integers (or rationals) when possible If the quadratic factors nicely, we look for two numbers \(p\) and \(q\) such that \[ p + q = 2b, \qquad p\cdot q = c. \] Then \[ x^2 + 2bx + c = (x + p)(x + q). \] Example \[ x^2 + 6x + 5 = (x + 1)(x + 5) \] because \(1 + 5 = 6\) and \(1\cdot 5 = 5\). When the numbers are not integers you can still use the quadratic formula on the equation \(x^2 + 2bx + c = 0\) and write the factored form \[ \bigl(x - r_1\bigr)\bigl(x - r_2\bigr) \] where \(r_1,r_2\) are the roots. Other common standard factoring patterns (for comparison) | Pattern | Identity | |---------|----------| | Difference of squares | \(a^2 - b^2 = (a-b)(a+b)\) | | Sum of cubes | \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\) | | Difference of cubes | \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\) | | Perfect-square trinomial | \(a^2 \pm 2ab + b^2 = (a \pm b)^2\) | Quick summary for \(x^2 + 2bx + c\) Check whether \(c = b^2\). If yes \(\rightarrow\) it is already \((x + b)^2\). Otherwise complete the square: \((x + b)^2 + (c - b^2)\). Or try to find two numbers that add to \(2b\) and multiply to \(c\). Would you like me to walk through several concrete numerical examples, show the same ideas for the general \(ax^2 + bx + c\), or move on to solving equations by factoring?19
Abbas Damerchi@abbasdamerchiOct 27youtube.comCeva Meets Pascal in A Beautiful Conic Interplay | Logic in Motion #20Two Cornerstone Results of Classical Geometry, Ceva & Pascal, Meeti...118
Abbas Damerchi@abbasdamerchiOct 27Just Published My New Short Course: #TCPDUMP youtube.com/playlist?list=… via @YouTubeyoutube.comTCPDUMP shorts | Security Researcher Pathtcpdump mastery in vertical Shorts — one flag per episode, terminal-first, step by step until you think like a security researcher. This collection walks you...430
