Answer:
C) -16; 0; -4 ± 2i
Step-by-step explanation:
The discriminant is a mathematical expression used in quadratic equations to determine the nature and number of real solutions.
[tex]\boxed{\begin{array}{l}\underline{\sf Discriminant}\\\\b^2-4ac\\\\\textsf{when $ax^2+bx+c=0$}\\\\\textsf{$b^2-4ac > 0 \implies$ two real solutions}\\\textsf{$b^2-4ac=0 \implies$ one real solution}\\\textsf{$b^2-4ac < 0 \implies$ no real solutions}\end{array}}[/tex]
Given quadratic equation:
[tex]x^2+8x+20=0[/tex]
Therefore, the coefficients are:
Substitute the values of a, b and c into the discriminant formula:
[tex]b^2-4ac=8^2-4(1)(20)\\\\b^2-4ac=64-4(20)\\\\b^2-4ac=64-80\\\\b^2-4ac=-16[/tex]
Therefore, the discriminant is -16.
As the discriminant of the given equation is less than zero, there are no real solutions.
To solve x² + 8x + 20 = 0, use the quadratic formula:
[tex]x=\dfrac{-8\pm\sqrt{-16}}{2(1)}\\\\\\x=\dfrac{-8\pm\sqrt{-16}}{2}\\\\\\x=\dfrac{-8\pm\sqrt{4^2\cdot -1}}{2}\\\\\\x=\dfrac{-8\pm\sqrt{4^2}\sqrt{-1}}{2}\\\\\\x=\dfrac{-8\pm4i}{2}\\\\\\x=-4\pm 2i[/tex]
Therefore, the two solutions are:
[tex]\textsf{Solution 1:}\quad x=-4 -2i\\\\\textsf{Solution 2:}\quad x=-4+2i[/tex]