Answer:
To complete the square and rewrite the quadratic function y = x² + 7x + 3 in vertex form, we follow these steps:
Factor out the coefficient of x² from the first two terms:
y = 1(x² + 7x) + 3
Take half of the coefficient of x (which is 7 in this case) and square it. Add this value inside the parentheses, and subtract the same value multiplied by the coefficient of x² (which is 1) outside the parentheses to maintain the same value of the expression:
y = 1(x² + 7x + (7/2)² - (7/2)²) + 3
Simplify inside the parentheses by combining the first three terms using the square of the binomial formula (a + b)² = a² + 2ab + b²:
y = 1(x + 7/2)² - 1/4 + 3
Combine the constant terms to simplify:
y = 1(x + 7/2)² + 11/4
Therefore, the quadratic function y = x² + 7x + 3 can be written in vertex form as y = (x + 7/2)² + 11/4. The vertex is located at the point (-7/2, 11/4).
Hope This Helps!