For what value of K will the equation 3x^2- 4x-(1+K)= 0 have equal roots?
To find the value of K for which the equation has equal roots, we can use the discriminant of the quadratic equation. The discriminant, denoted as Δ, is the part of the quadratic formula under the square root sign.
For a quadratic equation in the form ax^2 + bx + c = 0, the discriminant is given by the formula: Δ = b^2 - 4ac.
In this case, the given equation is 3x^2 - 4x - (1 + K) = 0.
Comparing it to the general form, we have:
a = 3, b = -4, c = -(1 + K).
Substituting these values into the discriminant formula, we get:
Δ = (-4)^2 - 4(3)(-(1 + K))
= 16 + 12(1 + K).
For the roots of a quadratic equation to be equal, the discriminant Δ must be equal to zero.
Setting Δ = 0, we have:
16 + 12(1 + K) = 0
Now, let us solve for K:
16 + 12(1 + K) = 0
16 + 12 + 12K = 0
12K + 28 = 0
12K = -28
K = -28/12
K = -7/3
Therefore, the value of K for which the equation 3x^2 - 4x - (1 + K) = 0 has equal roots is K = -7/3.