Three times the square of a certain positive number exceeds six times the number by nine. Find the number:
Read the words of the problem and convert it into an equation:
"Three times the square of a certain positive number" 3x^2
"Exceeds six times the number by nine" 6x + 9
These two expressions have to be equal: 3x^2 = 6x + 9
To solve we must set the quadratic equal to zero: 3x^2 - 6x - 9 = 0
We can factor out a 3: 3(x^2 - 2x - 3) = 0
Divide by 3: x^2 - 2x - 3 = 0
Factor to solve: (x - 3) (x + 1) = 0 x = 3 or x = -1
Since the original problem said the number was positive, "3" is the correct answer.
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