Answer:
The value of k for which the equation has two negative roots is k < 1.
Explanation:
To find the value of k for which the equation has two negative roots, we need to use the discriminant of the quadratic equation. The discriminant is the expression under the square root sign in the quadratic formula, and it determines the number and nature of the roots of the quadratic equation.
The quadratic equation in the picture is:
x^2 - 2(k - 3)x + 4k = 0
The discriminant of this equation is:
D = b^2 - 4ac
= (-2(k-3))^2 - 4(1)(4k)
= 4(k^2 - 10k + 9) - 16k
= 4k^2 - 56k + 36
For the equation to have two negative roots, the discriminant must be greater than zero and the coefficient of x^2 must be positive (since the leading coefficient is 1). So we have:
D > 0 and 4 > 0
Solving for k, we get:
4k^2 - 56k + 36 > 0
Dividing by 4 and simplifying, we get:
k^2 - 14k + 9 > 0
Factorizing the quadratic expression, we get:
(k - 1)(k - 13) > 0
The inequality is satisfied when either both factors are positive or both factors are negative. So we have two cases:
Case 1: (k - 1) > 0 and (k - 13) > 0
This gives us k > 13, which does not satisfy the condition that the roots should be negative.
Case 2: (k - 1) < 0 and (k - 13) < 0
This gives us k < 1, which satisfies the condition that the roots should be negative.
Therefore, the value of k for which the equation has two negative roots is k < 1.