Hi. I need help with these questions.
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Answer 11,12

11) k = -112) (i) 4, (ii) 6.5

Step-by-step explanation:

11.

Given equationx² + (k - 1)x - k = 0The roots are equalTo findThe value of kSolution

When the roots are equal, the discriminant equals zero

D = 0b² - 4ac = 0

Substituting the values and solving for k:

(k - 1)² - 4*1*(-k) = 0k² - 2k + 1 - 4k = 0k² + 2k + 1 = 0(k + 1)² = 0k + 1 = 0k = -1

12.

Given equation2x² - 6x + 1 = 0The roots are α and βTo find

The values of

(i) 2αβ² + 2α²β + 2αβ(ii) α² - 3αβ + β²Solution

The sum  and the product of the roots is:

α + β = - b/a = - ( - 6)/2 = 3αβ = c/a = 1/2

(i)

2αβ² + 2α²β + 2αβ =2αβ ( α + β + 1) = 2(1/2)(3 + 1) =1(4) =4

(ii)

α² - 3αβ + β² =(α + β)² - 5αβ = 3² - 5(1/2) =9 - 5/2 =6.5