Final answer:
To convert the trinomial 3x^2+6x+16 into a perfect square, the number needed is -13, which when adjusted for in the constant term, would allow the trinomial to be rewritten as a perfect square binomial squared.
Step-by-step explanation:
To convert the trinomial 3x^2+6x+16 into a perfect square, we can use the method of completing the square. Completing the square involves creating a binomial that when squared will produce the original quadratic trinomial. First, we need to focus on the x terms of the trinomial.
The coefficient of x^2 is 3, so our binomial will have the form (√x^2 + m)^2, where m is the number we are trying to find. For a perfect square trinomial, the middle term is twice the product of the square root of the a-coefficient and m. In our case, the middle term is 6x, and the square root of the a-coefficient is √3. As such, we have:
2 × √3 × m = 6x
√3 × m = 3x
m = 3x / √3
m = √3 × x
Now that we have m, we know that the binomial is (√3x + √3)^2, which results in:
3x² + 2√3 × 3x + 3
Since we initially had a constant term of 16, and our perfect square has a constant term of +3, the number needed to be added to 16 to make the trinomial a perfect square is:
Required number = Constant term of perfect square - Original constant term
Required number = 3 - 16
Required number = -13
Therefore, the number that allows the trinomial 3x^2+6x+16 to be rewritten into a perfect square is -13.