Question

Find the solutions to the given system of equations

Answer 1

$\bf \begin{cases} x^2+y^2=15\\ y^2=15-x^2\\ y=\sqrt{15-x^2}\\ -------\\ y=-x-6 \end{cases}\qquad \qquad \sqrt{15-x^2}=-x-6 \\\\\\ \textit{again, squaring both sides}\implies 15-x^2=(-x-6)^2\\\\ -------------------------------\\\\ 15-x^2=x^2+12x+36\implies 0=2x^2+12x+21 \\\\\\ \textit{now, let's check the discriminant of it} \qquad \begin{array}{llccll} 0=&{{ 2}}x^2&{{ +12}}x&{{ +21}}\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array}$

$\bf discriminant\implies b^2-4ac= \begin{cases} 0&\textit{one solution}\\ positive&\textit{two solutions}\\ negative&\textit{no solution} \end{cases} \\\\\\ 12^2-4(2)(21)\implies 144-168\implies -24$

notice, the discriminant is negative.


on the second system of equations there, check for any typos, there's a solution however, is none of those.

Answer 2

Your closeest awnsers D