To convert the quadratic function y=x^2+x+5 to vertex form, we can complete the square.
To complete the square, we start with the quadratic function in standard form: y = x^2 + x + 5
Step 1: Divide the coefficient of x by 2 and square the result.
1/2 * 1 = 1/4
(1/4)^2 = 1/16
Step 2: Add the result from step 1 to both sides of the equation.
y + 1/16 = x^2 + x + 5 + 1/16
Step 3: Rewrite the equation as the square of a binomial on the right side.
y + 1/16 = (x + 1/2)^2 + 20/16
Step 4: Simplify the right side.
y + 1/16 = (x + 1/2)^2 + 5/4
Step 5: Move the constant term to the left side.
y = (x + 1/2)^2 + 5/4 - 1/16
Step 6: Combine the constant terms on the right side.
y = (x + 1/2)^2 + 19/16
Therefore, the quadratic function y = x^2 + x + 5 in vertex form is y = (x + 1/2)^2 + 19/16.