one of the roots of the quadratic

equation x^2-(4+k)x+12=0
find,
1)the roots of the equation
2)the possible values of k

please show your workings

thanks in advance.

no root given

To find the roots of the quadratic equation and the possible values of k, we can use the quadratic formula. The formula is given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c = 0.

Let's find the roots first:

For the given equation x^2 - (4+k)x + 12 = 0, we can identify:

a = 1
b = -(4+k)
c = 12

Substitute these values into the quadratic formula:

x = (-(4+k) ± √((4+k)^2 - 4*1*12)) / (2*1)

Simplifying further:

x = (-(4+k) ± √((16+8k+k^2) - 48)) / 2

x = (-(4+k) ± √(16+8k+k^2-48)) / 2

Combining like terms:

x = (-(4+k) ± √(k^2+8k-32)) / 2

Now, let's simplify the expression under the square root:

k^2+8k-32 = (k+4)(k-8)

So, the expression becomes:

x = (-(4+k) ± √((k+4)(k-8))) / 2

Now, we have the roots of the equation:

Root 1: x1 = (-(4+k) + √((k+4)(k-8))) / 2
Root 2: x2 = (-(4+k) - √((k+4)(k-8))) / 2

To find the possible values of k, we need to consider where the roots are real. For the roots to be real, the expression inside the square root must be non-negative. So:

(k+4)(k-8) ≥ 0

To satisfy this inequality, either both factors must be positive or both must be negative:

1) If (k+4) > 0 and (k-8) > 0, then k > -4 and k > 8. Therefore, k > 8.

2) If (k+4) < 0 and (k-8) < 0, then k < -4 and k < 8. Therefore, k < -4.

So, the possible values of k are k < -4 or k > 8.

Therefore, the roots of the equation and the possible values of k are:

1) The roots are:
- Root 1: x1 = (-(4+k) + √((k+4)(k-8))) / 2
- Root 2: x2 = (-(4+k) - √((k+4)(k-8))) / 2

2) The possible values of k are: k < -4 or k > 8.