For this case we must solve the following questions:Question 1:We must solve the following quadratic equation:[tex]x ^ 2 + 1 = 0[/tex]We subtract 1 from both sides of the equation:[tex]x ^ 2 + 1-1 = -1\\x ^ 2 = -1[/tex]We apply square root to eliminate the exponent:[tex]x = \pm \sqrt {-1}[/tex]The solution is not real.[tex]x_ {1} = + i\\x_ {2} = - i[/tex]Answer:[tex]x_ {1} = + i\\x_ {2} = - i[/tex]Option DQuestion 2:We must solve the following quadratic equation:[tex]x ^ 2-10x + 24 = 0[/tex]We factor the equation, that is, we must look for two numbers that when multiplied give as result 24 and when summed with -10. These numbers are -6 and -4:[tex]-6-4 = -10\\-6 * -4 = 24\\(x-6) (x-4) = 0[/tex]The roots are:[tex]x_ {1} = 6\\x_ {2} = 4[/tex]ANswer:[tex]x_ {1} = 6\\x_ {2} = 4[/tex]Option DQuestion 3:We must complete squares of the following expression:[tex]x ^ 2-18x = 7[/tex]We add [tex](\frac {b} {2}) ^ 2 = (\frac {-18} {2}) ^ 2 = (- 9) ^ 2[/tex]Then:[tex]x ^ 2-18x + (- 9) ^ 2 = 7 + (- 9) ^ 2\\x ^ 2-18x + (- 9) ^ 2 = 7 + 81\\x ^ 2-18x + (- 9) ^ 2 = 88\\(x-9) ^ 2 = 88[/tex]Then, to complete the square, add it to both sides [tex](-9) ^ 2 = 81[/tex]Answer:Option C