Question

Factor the expression c^2+4c-4

Answer 1

$Method\ 1^o\\Use\ a^2+2ab+b^2=(a+b)^2\\\\c^2+4c-4=0\ \ \ |add\ 8\ to\ both\ sides\\\\c^2+2c\cdot2+4=8\\\\c^2+2c\cdot2+2^2=8\\\\(c+2)^2=8\iff c+2=-\sqrt8\ or\ c+2=\sqrt8\\\\/\sqrt8=\sqrt{4\cdot2}=\sqrt4\cdot\sqrt2=2\sqrt2/\\\\c+2=-2\sqrt2\ or\ c+2=2\sqrt2\ \ \ |subtract\ 2\ from\ both\ sides\\\\\boxed{c=-2\sqrt2-2\ or\ c=2\sqrt2-2}$

$Method\ 2^o:\\change\ "c"\ to\ "x"\\\\x^2+4x-4=0\\a=1;\ b=4;\ c=-4\\\\\Delta=b^2-4ac\to\Delta=4^2-4\cdot1\cdot(-4)=16+16=32 \ \textgreater \ 0\\\\then\ x_1=\dfrac{-b-\sqrt\Delta}{2a}\ and\ x_2=\dfrac{-b+\sqrt\Delta}{2a}\\\\\sqrt\Delta=\sqrt{32}=\sqrt{16\cdot2}=\sqrt{16}\cdot\sqrt2=4\sqrt2\\\\x_1=\dfrac{-4-4\sqrt2}{2\cdot1}=\dfrac{-4-4\sqrt2}{2}=-2-2\sqrt2\\\\x_2=\dfrac{-4+4\sqrt2}{2\cdot1}=\dfrac{-4+4\sqrt2}{2}=-2+2\sqrt2\\\\\boxed{c=-2-2\sqrt2\ or\ c=-2+2\sqrt2}$