Answer:
1. x = 7 or x = -2
2. x = 3/5 or x = -3
3. x = 3/2 or x = -2/3
Explanation:
1. Solve for x over the real numbers:
x^2 - 5 x - 14 = 0
The left hand side factors into a product with two terms:
(x - 7) (x + 2) = 0
Split into two equations:
x - 7 = 0 or x + 2 = 0
Add 7 to both sides:
x = 7 or x + 2 = 0
Subtract 2 from both sides:
Answer: x = 7 or x = -2
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2> Solve for x over the real numbers:
5 x^2 + 12 x = 9
Divide both sides by 5:
x^2 + (12 x)/5 = 9/5
Add 36/25 to both sides:
x^2 + (12 x)/5 + 36/25 = 81/25
Write the left hand side as a square:
(x + 6/5)^2 = 81/25
Take the square root of both sides:
x + 6/5 = 9/5 or x + 6/5 = -9/5
Subtract 6/5 from both sides:
x = 3/5 or x + 6/5 = -9/5
Subtract 6/5 from both sides:
Answer: x = 3/5 or x = -3
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3. Solve for x over the real numbers:
6 x^2 = 5 x + 6
Subtract 5 x + 6 from both sides:
6 x^2 - 5 x - 6 = 0
The left hand side factors into a product with two terms:
(2 x - 3) (3 x + 2) = 0
Split into two equations:
2 x - 3 = 0 or 3 x + 2 = 0
Add 3 to both sides:
2 x = 3 or 3 x + 2 = 0
Divide both sides by 2:
x = 3/2 or 3 x + 2 = 0
Subtract 2 from both sides:
x = 3/2 or 3 x = -2
Divide both sides by 3:
Answer: x = 3/2 or x = -2/3