Hi. Please help me with these questions
See image for question. Answer no 6 and 7

6. See solution7. k = 2, k = -6

Step-by-step explanation:

6.

Given equation:

2(x + 2)² + p(x + 1) = 0The sum of the roots is q1 and the product of the roots is q2

Need to show that q1+q2 = 0

Solution

Bringing the equation into standard form of ax² + bx + c = 0:

2(x + 2)² + p(x + 1) = 0 2x² + 8x + 8 + px + p = 02x² + (p + 8)x + (p + 8) = 0

Sum of the roots:

q1 = - b/a = -(p + 8)/2

Product of the roots:

q2 = c/a = (p + 8)/2

We see that q1 and q2 are opposite numbers, therefore their sum equals zero:

q1 + q2 = -(p + 8)/2 +  (p + 8)/2 = 0

7.

Given quadratic equation:

x² - (k + 2)x + 4 = 0It has equal roots

Need to find the possible values of k

Solution

When the quadratic equation has equal roots, then its discriminant is equal to zero:

D = 0√b² - 4ac = 0√(-k -2)² - 4*1*4 = 0√k² + 4k + 4 - 16 = 0√k² + 4k - 12 = 0k² + 4k - 12 = 0k = {-4 ± √4² -4*1*(-12)}/2 k = (-4 ± √16 + 48)/2k = (-4±√64)/2k = -2 ± 4k = 2, k = -6