Question

Hi. I need help with these questions.
See age for question.
Answer 11,12

Answer 1

• 11) k = -1
• 12) (i) 4, (ii) 6.5

Explanation:

11. ......................................................

Given equation

• x² + (k - 1)x - k = 0
• The roots are equal

To find

• The value of k

Solution

When the roots are equal, the discriminant equals zero

• D = 0
• b² - 4ac = 0

Substituting the values and solving for k:

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

12. ......................................................

Given equation

• 2x² - 6x + 1 = 0
• The roots are α and β

To find

The values of

• (i) 2αβ² + 2α²β + 2αβ
• (ii) α² - 3αβ + β²

Solution

The sum and the product of the roots is:

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