Find the exact values of k for which kx^2- kx - 2x - 3 = 0 has equal roots
To find the exact values of k for which the equation has equal roots, we can use the discriminant.
The discriminant of a quadratic equation, ax^2 + bx + c = 0, is given by b^2 - 4ac. If the discriminant is equal to zero, the quadratic equation has equal roots.
In the given equation, kx^2 - kx - 2x - 3 = 0, the quadratic term coefficient is k, the linear term coefficient is (-k - 2), and the constant term is -3.
Substituting these values into the discriminant formula, we get:
(-k - 2)^2 - 4(k)(-3)
Expanding the equation gives us:
k^2 + 4k + 4 - 4k + 24
Simplifying further:
k^2 + 24 + 4
Combining like terms, we have:
k^2 + 28
Since we want the equation to have equal roots, the discriminant should be equal to zero.
Setting the discriminant equal to zero, we have:
k^2 + 28 = 0
Rearranging the equation:
k^2 = -28
Taking the square root of both sides, we get:
k = ±√(-28)
Since the square root of a negative number is not a real number, there are no real values of k for which the equation has equal roots.
To find the values of k for which the equation kx^2 - kx - 2x - 3 = 0 has equal roots, we can use the concept of discriminant.
The discriminant (D) is given by the formula D = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
For the given equation kx^2 - kx - 2x - 3 = 0, let's compare the coefficients with the standard form ax^2 + bx + c = 0:
a = k
b = -k - 2
c = -3
Now, we can apply the discriminant to get the conditions for equal roots.
For equal roots, D = 0.
So, we have:
D = (-k - 2)^2 - 4(k)(-3) = 0
Expanding and simplifying the equation:
k^2 + 4k + 4 - 12k = 0
k^2 - 8k + 4 = 0
To solve this quadratic equation for k, we can either factorize it or use the quadratic formula.
Factoring the equation:
(k - 2)(k - 2) = 0
(k - 2)^2 = 0
Taking the square root of both sides:
k - 2 = 0
k = 2
So, the equation kx^2 - kx - 2x - 3 = 0 has equal roots when k = 2.