Question

Dertermine the positive v alue of p for which the equations $x^2+2px+64=0and{X}^2-8x+2p=0$ will both have real roots.

Answer 1

$D\ge 0$ for real roots

where $D=b^2-4ac$

$D_1=4p^2-4\left(64\right)\ge 0$

$4(p^2-64)\ge 0=p^2\ge 64$

$\Rightarrow p\ge 8(p\ge 0)$

$D_2=64-8p\ge 0$

$=8p\le 64\Rightarrow p\le 8$

$\Rightarrow p=8$