for what are value of k are the roots of 3x^2-6x+k=0 equal?
For the roots of the quadratic equation
ax^2+bx+c=0
to be equal, the discriminant Δ must equal to zero, where
Δ=b^2-4ac...(1)
For the given equation
3x^2-6x+k=0
a=3,b=-6, c=k
Substitute a,b,and c in (1), equate to zero and solve for k.
Post your answer for checking if you wish.
To find the values of k for which the roots of the quadratic equation 3x^2 - 6x + k = 0 are equal, we need to look at the discriminant of the quadratic equation.
The discriminant, denoted as Δ, is the expression under the square root in the quadratic formula (x = [-b ± √(b^2 - 4ac)] / 2a), and it determines the nature of the roots.
For a quadratic equation of the form ax^2 + bx + c = 0, the discriminant can be calculated as Δ = b^2 - 4ac.
In this case, the quadratic equation is 3x^2 - 6x + k = 0, so a = 3, b = -6, and c = k.
The discriminant can be written as Δ = (-6)^2 - 4(3)(k) = 36 - 12k.
For the roots to be equal, the discriminant must be equal to zero, since in that case, the ± sign in the quadratic formula disappears, and we are left with a single root.
Setting Δ equal to zero and solving for k will give us the values where the roots are equal:
36 - 12k = 0
-12k = -36
k = 3
Therefore, the value of k for which the roots of the quadratic equation 3x^2 - 6x + k = 0 are equal is k = 3.