The sum of two numbers is 12, their product is 96. Compute these two numbers. Explain.​

Answer:The numbers are[tex]6+2\sqrt{15}i[/tex]   and  [tex]6-2\sqrt{15}i[/tex]Step-by-step explanation:Letx and y -----> the numberswe know that[tex]x+y=12[/tex] -----> [tex]y=12-x[/tex] ------> equation A[tex]xy=96[/tex] ----> equation Bsubstitute equation A in equation B and solve for x[tex]x(12-x)=96\\12x-x^{2}=96\\x^{2} -12x+96=0[/tex]Solve the quadratic equation The formula to solve a quadratic equation of the form [tex]ax^{2} +bx+c=0[/tex] is equal to[tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex]in this problem we have[tex]x^{2} -12x+96=0[/tex]  so[tex]a=1\\b=-12\\c=96[/tex]substitute[tex]x=\frac{-(-12)(+/-)\sqrt{-12^{2}-4(1)(96)}} {2(1)}[/tex][tex]x=\frac{12(+/-)\sqrt{-240}} {2}[/tex]Remember that[tex]i^{2}=\sqrt{-1}[/tex][tex]x=\frac{12(+/-)\sqrt{240}i} {2}[/tex][tex]x=\frac{12(+/-)4\sqrt{15}i} {2}[/tex]Simplify[tex]x=6(+/-)2\sqrt{15}i[/tex][tex]x1=6+2\sqrt{15}i[/tex][tex]x2=6-2\sqrt{15}i[/tex]we have two solutionsFind the value of y for the first solutionFor [tex]x1=6+2\sqrt{15}i[/tex] [tex]y=12-x[/tex]substitute [tex]y1=12-(6+2\sqrt{15}i)[/tex] [tex]y1=6-2\sqrt{15}i[/tex]Find the value of y for the second solutionFor [tex]x2=6-2\sqrt{15}i[/tex] [tex]y2=12-x[/tex]substitute [tex]y2=12-(6-2\sqrt{15}i)[/tex] [tex]y2=6+2\sqrt{15}i[/tex]thereforeThe numbers are[tex]6+2\sqrt{15}i[/tex]   and  [tex]6-2\sqrt{15}i[/tex]