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Answer:

To find the sum of all possible non-negative integral values of k for which the roots of the equation 4x² - 3kx + 2k = 0 are real and |k| ≤ 13, we need to use the discriminant of the quadratic equation.

The quadratic equation in standard form is ax² + bx + c = 0, where a = 4, b = -3k, and c = 2k.

For the roots to be real, the discriminant (D) should be greater than or equal to zero:

D = b² - 4ac

Substituting the values:

D = (-3k)² - 4 * 4 * 2k

D = 9k² - 32k

Now, the discriminant (D) should be greater than or equal to zero:

9k² - 32k ≥ 0

To find the possible values of k, we need to solve this inequality:

9k² - 32k ≥ 0

k(9k - 32) ≥ 0

The values of k for which the inequality holds true are:

k ≤ 0 (non-negative integral values)

9k - 32 ≥ 0

k ≥ 32/9 ≈ 3.56

However, the constraint given is |k| ≤ 13, so the valid range for k is -13 ≤ k ≤ 13.

Now, let's find the sum of all possible non-negative integral values of k within this valid range:

Possible non-negative integral values of k: 0, 1, 2, 3, 4, ..., 13

The sum of these values is the sum of the first 14 natural numbers:

Sum = (14 * 15) / 2 = 105

Thus, the sum of all possible non-negative integral values of k is 105.

Step-by-step explanation:

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To find the sum of all possible non-negative

integral values of k for which the roots of the

equation 4x² - 3kx + 2k = 0 are real and Ikl

≤ 13, we need to use the discriminant of the

quadratic equation.

The quadratic equation in standard form is

ax² + bx + c = 0, where a = 4, b = -3k, and c =

2k.

For the roots to be real, the discriminant (D)

should be greater than or equal to zero:

D = b² - 4ac

Substituting the values:

Now, the discriminant (D) should be greater

than or equal to zero:

9k² -32k≥0

To find the possible values of k, we need to

solve this inequality:

9k² -32k ≥ 0

k(9k - 32) ≥ 0

The values of k for which the inequality

holds true are:

k≤ 0 (non-negative integral values)

9k - 32 ≥ 0

k≥ 32/9 = 3.56

the valid range for k is -13 ≤ k ≤ 13.

Now, let's find the sum of all possible

non-negative integral values of k within this

valid range:

Possible non-negative integral values of k: 0,

1, 2, 3, 4, ..., 13

The sum of these values is the sum of the

first 14 natural numbers:

Sum = (14*15) / 2 = 105

Thus, the sum of all possible non-negative

integral values of k is 105.