Question

Please help; i’m confused

Answer 1

The values of k that make the given equation having imaginary roots are:

k < -18, or (-∞,-18).

Explanation:

Nature of the Roots of a Quadratic Equation

The standard representation of a quadratic equation is:

`ax^2+bx+c=0`

where a,b, and c are constants.

Solving with the quadratic formula:

`\displaystyle x=(-b\pm √(b^2-4ac))/(2a)`

The expression:

`d=b^2-4ac`

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:

`-2x^2+12x+k=0`

Comparing with the standard quadratic equation, we have:

a=-2, b=12, c=k

Calculating the discriminant:

`d=12^2-4(-2)k`

`d=144+8k`

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).