How do you write #y=x^2+12x+32# into vertex form?

The form of the vertex is:

So:

To write the quadratic function ( y = x^2 + 12x + 32 ) into vertex form, follow these steps:

1. Complete the square on the quadratic term.
2. Factor out any constants outside of the square term.
3. Write the expression in vertex form.

Completing the square: [ y = x^2 + 12x + 32 ] [ y = (x^2 + 12x + \underline{\hspace{0.3cm}}) + 32 - \underline{\hspace{0.3cm}} ]

To complete the square, take half of the coefficient of (x) (which is (12/2 = 6)), square it ((6^2 = 36)), and add it inside the parentheses:

[ y = (x^2 + 12x + 36) - 36 + 32 ]

Factor out the constant outside of the parentheses:

[ y = (x^2 + 12x + 36) - 4 ]

Rewrite the expression in vertex form by factoring the squared term:

[ y = (x + 6)^2 - 4 ]

So, the quadratic function ( y = x^2 + 12x + 32 ) in vertex form is ( y = (x + 6)^2 - 4 ).