Question

Determine o conjunto solução da equação (x - 2p)(x + 5p) = 8p^2 na incógnita X.

Answer 1

$(x-2p)(x+5p)=8p^{2}\\x^{2}+5xp-2xp-10p^{2}=8p^{2}\\x^{2}+3xp-10p^{2}-8p^{2}=0\\x^{2}+(3p)x-18p^{2}=0\\\\\Delta=b^{2}-4ac\\\Delta=(3p)^{2}-4\cdot1\cdot(-18p^{2})\\\Delta=9p^{2}+72p^{2}\\\Delta=81p^{2}\\\sqrt{\Delta}=\sqrt{81}\sqrt{p^{2}}\\\sqrt{\Delta}=9p\\\\x=\dfrac{-b\pm\sqrt{\Delta}}{2a}=\dfrac{-3p\pm9p}{2\cdot1}=\dfrac{-3p\pm9p}{2}$

Então, temos duas possibilidades para x:

$x=\dfrac{-3p+9p}{2}=\dfrac{6p}{2}~~~~~~~~\therefore~~~~\boxed{\boxed{x=3p}}\\\\\\x=\dfrac{-3p-9p}{2}=-\dfrac{12p}{2}~~~~\therefore~~~~\boxed{\boxed{x=-6p}}$