Final answer:
To complete the square for y = 5x² + 15x + 4, rewrite the function as y = 5(x + 3/2)² - 29/4. The vertex of the parabola formed by this function is at (-3/2, -29/4).
Step-by-step explanation:
To complete the square for the function y = 5x² + 15x + 4, first divide the quadratic and linear coefficients by the leading coefficient, if it is not 1. In this case, we divide by 5, giving us y = x² + 3x. We then take half of the linear coefficient (3/2), square it, add it inside the parentheses, and subtract it outside (to keep the equation balanced, we also multiply this term by the leading coefficient, 5).
The completed square form of the equation is y = 5(x + 3/2)² - 5(3/2)² + 4. Simplify the constant terms to find the completed square: y = 5(x + 3/2)² - 45/4 + 16/4, which simplifies further to y = 5(x + 3/2)² - 29/4.
The vertex of the parabola is at the point where x = -3/2 and y = -29/4, since the vertex form of a quadratic function is y = a(x - h)² + k, where (h, k) is the vertex.