Question

Consider the quadratic equation

x squared +5x+11=0

A: What is the discriminant of the quadratic equation?
B: Based on the discriminant, which statement about the number and type of solutions to the equation is correct?

Select one answer for part A, and select one answer for part B.

A: -19
B: The equation has one repeated real solution.
B: The equation has two real solutions.
A: 0
B: The equation has two complex solutions.
A: 69

Answer 1

`\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}`

Answer 2

A: -19

B: The equation has two complex solutions.

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.