Answer:
B) -64; 0; -1 ± 4i
Step-by-step explanation:
The discriminant of a quadratic equation is a mathematical expression derived from its coefficients, indicating the nature and number of its 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+2x+17=0[/tex]
Therefore, the coefficients are:
Substitute the values of a, b and c into the discriminant formula:
[tex]b^2-4ac=2^2-4(1)(17)\\\\b^2-4ac=4-4(17)\\\\b^2-4ac=4-68\\\\b^2-4ac=-64[/tex]
Therefore, the discriminant is -64.
As the discriminant of the given equation is less than zero, there are no real solutions.
To solve x² + 2x + 17 = 0, use the quadratic formula:
[tex]x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\\\x=\dfrac{-2\pm\sqrt{-64}}{2(1)}\\\\\\x=\dfrac{-2\pm\sqrt{-64}}{2}\\\\\\x=\dfrac{-2\pm\sqrt{8^2\cdot -1}}{2}\\\\\\x=\dfrac{-2\pm\sqrt{8^2}\sqrt{-1}}{2}\\\\\\x=\dfrac{-2\pm 8i}{2}\\\\\\x=-1\pm4i[/tex]
Therefore, the two solutions are:
[tex]\textsf{Solution 1:}\quad x=-1-4i\\\\\textsf{Solution 2:}\quad x=-1+4i[/tex]
