Final answer:
The equation f(x) = 2x² - 4x + 5 can be written in vertex form as f(x) = 2(x - 1)² + 3.
Step-by-step explanation:
The given equation, f(x) = 2x² - 4x + 5, is already in standard form. To convert it to vertex form, we can use the completing the square method.
First, let's factor out the common factor of 2 from the first two terms to get:
f(x) = 2(x² - 2x)+ 5
Next, we need to complete the square inside the parentheses. To do this, we take half of the coefficient of the x-term (-2/2 = -1) and square it (-1) to get 1.
Now, add and subtract 1 inside the parentheses:
f(x) = 2(x² - 2x + 1 - 1) + 5
By factoring the perfect square trinomial, we can rewrite the equation as:
f(x) = 2((x - 1)² - 1) + 5
Finally, simplify:
f(x) = 2(x - 1)² - 2 + 5
Therefore, the equation f(x) = 2x² - 4x + 5 written in vertex form is f(x) = 2(x - 1)² + 3.