Question

Consider the quadratic equation

Answer 1

Answer:

A: -19

B: The equation has two complex solutions.

Step-by-step explanation:

(A) Compare

... x² +5x +11 = 0

to the form

... ax² + bx + c = 0

and you see that a=1, b=5, c=11.

The discriminat (d) is computed as

... d = b²-4ac

Putting the above values in this equation for a, b, c, we get

... d = 5² -4·1·11 = 25 -44 = -19

_____

(B) The solutions are ...

... x = (-b ±√d)/(2a) = (-5 ±√-19)/2

The square root of -19 is imaginary, so there are two complex solutions. It will be the case that the two solutions are complex whenever the discriminant is negative.

Answer 2

$\bf \qquad \qquad \qquad \textit{discriminant of a quadratic} \\\\\\ \stackrel{\stackrel{a}{\downarrow }}{1}x^2\stackrel{\stackrel{b}{\downarrow }}{+5}x\stackrel{\stackrel{c}{\downarrow }}{+11}=0 ~~~~~~~~ \stackrel{discriminant}{b^2-4ac}= \begin{cases} 0&\textit{one real solution}\\ positive&\textit{two real solutions}\\ negative&\textit{no real solution}\\ &\boxed{\textit{two complex solutions}} \end{cases} \\\\\\ (5)^2-4(1)(11)\implies 25-44\implies -19\impliedby \boxed{negative}$