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Answer:

To find the value of k for which the equation x² + k(2x + k - 1) = 0 has real and equal roots, we need to use the discriminant of the quadratic equation.

The discriminant (Δ) of a quadratic equation ax² + bx + c = 0 is given by Δ = b² - 4ac.

For the given equation, a = 1, b = 2k, and c = k - 1.

Now, the condition for real and equal roots is when the discriminant is equal to zero, i.e., Δ = 0.

So, let's set the discriminant to zero and solve for k:

Δ = (2k)² - 4(1)(k - 1) = 4k² - 4(k - 1) = 0

Simplifying further:

4k² - 4k + 4 = 0

Dividing the entire equation by 4:

k² - k + 1 = 0

Now, to find the value of k, we can use the quadratic formula:

The quadratic formula is given by x = (-b ± √(b² - 4ac)) / 2a.

For our equation, a = 1, b = -1, and c = 1:

k = [1 ± √((-1)² - 4(1)(1))] / 2(1)

k = [1 ± √(1 - 4)] / 2

k = [1 ± √(-3)] / 2

Since the discriminant is negative (Δ = -3), the equation has no real roots, and hence, there is no real value of k for which the equation x² + k(2x + k - 1) = 0 has real and equal roots.

Step-by-step explanation:

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