To rewrite the quadratic function f(x)=2x² +12x+16 in standard form, we need to complete the square. The standard form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola.
First, factor out the coefficient of x² from the first two terms: f(x) = 2(x² + 6x) + 16.
Find the number that completes the square for x² + 6x.
This number is (6/2)² = 9.
Add and subtract 9 inside the parenthesis: f(x) = 2[(x² + 6x + 9) - 9] + 16.
Now, factor the perfect square trinomial and simplify: f(x) = 2[(x + 3)²] - 2(9) + 16.
Finally, f(x) = 2(x + 3)² + 16 - 18, which simplifies to f(x) = 2(x + 3)² - 2.
This is the quadratic function in standard form.