Answer:
-14.5
Explanation:
We can use the given relation to write an equation for the number. The extraneous solution must be ignored.
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setup
Let x represent the number. Then 3 less than twice the number is (2x -3). The product is 464:
x(2x -3) = 464
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solution
2x^2 -3x = 464 . . . . eliminate parentheses
x^2 -3/2x = 232 . . . . divide by 2
x^2 -3/2x +9/16 = 232 +9/16 . . . . "complete the square"
(x -3/4)^2 = 232.5625 . . . . . write the trinomial as a square
x -3/4 = ±√232.5625 = ±15.25 . . . . take the square root
x = 0.75 -15.25 = -14.5 . . . . ignore the positive root
The number is -14.5.
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Additional comment
A perfect square trinomial is of the form ...
(x +a)² = x² +2ax +a²
That is, the constant (a²) is the square of half the x-coefficient: (2a/2)² = a². We "complete the square" by making sure the trinomial is a perfect square trinomial. In this case, it means adding a² = (3/4)² to both sides of the equation.