Final answer:
The vertex form of the quadratic equation y = 2x² - 20x + 16 is found by completing the square process, resulting in y = 2(x - 5)² - 34 with the vertex at (5, -34).
Step-by-step explanation:
To find the vertex form of the quadratic equation y = 2x² - 20x + 16, we use the process of completing the square. The vertex form of a quadratic equation is given by y = a(x - h)² + k, where (h, k) is the vertex of the parabola. Here's how to transform the given equation step-by-step:
- Start with the original equation: y = 2x² - 20x + 16.
- Factor out the coefficient of the x² term: y = 2(x² - 10x) + 16.
- Complete the square for the expression in parentheses: Find the term that completes the square, which is (10/2)² = 25, and add and subtract it inside the parentheses: y = 2(x² - 10x + 25 - 25) + 16.
- Add and subtract 25 inside the parentheses and simplify: y = 2((x - 5)² - 25) + 16.
- Distribute the 2 and combine like terms: y = 2(x - 5)² - 50 + 16.
- Finally, add -50 and 16 to combine the constants: y = 2(x - 5)² - 34.
The vertex form of the equation is y = 2(x - 5)² - 34, with the vertex being at (5, -34).