Answer:
a = -1/2 & k = 3/7
Step-by-step explanation:
Solve for a over the real numbers:
24 a^3 + 8 a^2 + 6 a + 4 = 0
The left hand side factors into a product with three terms:
2 (2 a + 1) (6 a^2 - a + 2) = 0
Divide both sides by 2:
(2 a + 1) (6 a^2 - a + 2) = 0
Split into two equations:
2 a + 1 = 0 or 6 a^2 - a + 2 = 0
Subtract 1 from both sides:
2 a = -1 or 6 a^2 - a + 2 = 0
Divide both sides by 2:
a = -1/2 or 6 a^2 - a + 2 = 0
Divide both sides by 6:
a = -1/2 or a^2 - a/6 + 1/3 = 0
Subtract 1/3 from both sides:
a = -1/2 or a^2 - a/6 = -1/3
Add 1/144 to both sides:
a = -1/2 or a^2 - a/6 + 1/144 = -47/144
Write the left hand side as a square:
a = -1/2 or (a - 1/12)^2 = -47/144
(a - 1/12)^2 = -47/144 has no solution since for all a on the real line, (a - 1/12)^2 >=0 and -47/144<0:
Answer: a = -1/2
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Solve for k over the real numbers:
(7 k - 3) (k^2 - 2 k + 7) = 0
Split into two equations:
7 k - 3 = 0 or k^2 - 2 k + 7 = 0
Add 3 to both sides:
7 k = 3 or k^2 - 2 k + 7 = 0
Divide both sides by 7:
k = 3/7 or k^2 - 2 k + 7 = 0
Subtract 7 from both sides:
k = 3/7 or k^2 - 2 k = -7
Add 1 to both sides:
k = 3/7 or k^2 - 2 k + 1 = -6
Write the left hand side as a square:
k = 3/7 or (k - 1)^2 = -6
(k - 1)^2 = -6 has no solution since for all k on the real line, (k - 1)^2 >=0 and -6<0:
Answer: k = 3/7
