Answer:
The values of k that make the given equation having imaginary roots are:
k < -18, or (-∞,-18).
Step-by-step explanation:
Nature of the Roots of a Quadratic Equation
The standard representation of a quadratic equation is:
[tex]ax^2+bx+c=0[/tex]
where a,b, and c are constants.
Solving with the quadratic formula:
[tex]\displaystyle x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
The expression:
[tex]d=b^2-4ac[/tex]
Is called the discriminant. The discriminant determines the nature of the roots of a quadratic equation as follows:
If d=0, there is only one real root.
if d>0, there are two different real roots
if d<0, there are two different imaginary (complex) roots
We are given the equation:
[tex]-2x^2+12x+k=0[/tex]
Comparing with the standard quadratic equation, we have:
a=-2, b=12, c=k
Calculating the discriminant:
[tex]d=12^2-4(-2)k[/tex]
[tex]d=144+8k[/tex]
If the equation has imaginary roots, then d<0, thus:
144 + 8k < 0
Subtracting 144:
8k < -144
Dividing by 8:
k < -18
The values of k that make the given equation having imaginary roots are: k < -18, or (-∞,-18).
