Question

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

Answer 1

• 6. See solution
• 7. k = 2, k = -6

Explanation:

6.

Given equation:

• 2(x + 2)² + p(x + 1) = 0

• The 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 = 0
• 2x² + (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

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7.

Given quadratic equation:

• x² - (k + 2)x + 4 = 0

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