The first step in rewriting the quadratic equation y = 4x² + 2x + 7 in vertex form y = a(x – h)² + k is to complete the square.
To complete the square, you need to focus on the quadratic and linear terms, which are 4x² + 2x. Here’s how you can do it:
- Factor out the leading coefficient (4) from the terms 4x² and 2x:
- This gives you: y = 4(x² + (2/4)x) + 7
- Now, simplify the linear term inside the parentheses: y = 4(x² + (1/2)x) + 7
- Next, to complete the square, take half of the coefficient of x (which is 1/2), square it (which gives 1/4), and add and subtract it inside the parentheses:
- This leads you to: y = 4(x² + (1/2)x + 1/4 – 1/4) + 7
- Now, restructure the equation: y = 4((x + 1/4)² – 1/4) + 7
- Distribute the 4: y = 4(x + 1/4)² – 1 + 7
- Simplify: y = 4(x + 1/4)² + 6
After completing the square, you can clearly see the vertex form as y = 4(x – (-1/4))² + 6 with a = 4, h = -1/4, and k = 6.