one of the roots of the quadratic equation x^2-(4+k)x+12=0
fine,
1)the roots of the equation
2)the possible values of k
please show your workings
thanks in advance.
x^2-(4+k)x = -12
x^2 -(4+k)x +(4+k)^2/4=-12 + (4+k)^2/4
take sqrt of each side
(x -(4+k)/2 ) =sqrt[ (4+k)^2/4-12]
ok, for the real roots, then
(4+k)^2/4-12>=0 or
(4+k)^2/4>=12 or
4+k>=sqrt24
k>=4+2 sqrt6
so k is greater than 4+2sqrt6, but sqrt 6 can be two values, or
k has to be between 4-2sqrt6 and 4+2sqrt 6
huh, thanks bro, but i didn't comprehen your workings.
How did you get (4+k)^2/4
read up on the discriminant of a quadratic: b^2-4ac
To find the roots of the given quadratic equation x^2 - (4+k)x + 12 = 0 and the possible values of k, we can make use of the quadratic formula.
The quadratic formula states that for a quadratic equation in the form ax^2 + bx + c = 0, the roots can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Comparing this with our equation x^2 - (4+k)x + 12 = 0, we can identify a = 1, b = -(4+k), and c = 12.
Now, let's find the roots of the equation by applying the quadratic formula:
x = (-(4+k) ± √((4+k)^2 - 4(1)(12))) / (2(1))
Simplifying this further:
x = (-(4+k) ± √(16 + 8k + k^2 - 48)) / 2
x = (-(4+k) ± √(k^2 + 8k - 32)) / 2
Now, to find the roots, we need the discriminant (the value inside the square root) to be greater than or equal to 0:
k^2 + 8k - 32 ≥ 0
To solve this quadratic inequality, we can factorize it or use the quadratic formula again. Factoring this inequality, we get:
(k - 4)(k + 8) ≥ 0
Now, we can consider the following cases:
1) When (k - 4) ≥ 0 and (k + 8) ≥ 0:
This satisfies the condition when k ≥ 4 and k ≥ -8, which means k ≥ 4.
2) When (k - 4) ≤ 0 and (k + 8) ≤ 0:
This satisfies the condition when k ≤ 4 and k ≤ -8, which means k ≤ -8.
Thus, the possible values of k are k ≥ 4 and k ≤ -8.
Finally, substituting these values of k back into the quadratic formula, we can find the roots of the equation:
x = (-(4+k) ± √(k^2 + 8k - 32)) / 2
Plug in k = 4:
x = (-(4+4) ± √((4^2) + 8(4) - 32)) / 2
x = (-8 ± √(16 + 32 - 32)) / 2
x = (-8 ± √16) / 2
x = (-8 ± 4) / 2
So, the roots when k = 4 are:
x₁ = (-8 + 4) / 2 = -2
x₂ = (-8 - 4) / 2 = -6
Plug in k = -8:
x = (-(4+(-8)) ± √((-8)^2 + 8(-8) - 32)) / 2
x = (-4 ± √(64 - 64 - 32)) / 2
x = (-4 ± √(-32)) / 2
As the discriminant is negative, there are no real roots for k = -8.
Therefore, the roots of the equation are x = -2 and x = -6, and the possible values of k are k ≥ 4.