Answer:
To factor the quadratic function f(x) = x² + 6x - 16, we need to find two integers whose product is -16 and whose sum is 6. The pair of integers that satisfies these conditions is -8 and 2.
We can then factor the quadratic as follows:
f(x) = x² + 6x - 16
f(x) = (x - 8)(x + 2)
The roots of the quadratic are the values of x that make the quadratic equal to 0. In this case, the roots are x = 8 and x = -2.
The vertex of a quadratic function is the point on the graph where the function reaches its minimum or maximum value. The vertex of a quadratic of the form f(x) = ax² + bx + c is given by the coordinates (-b/2a, f(-b/2a)).
In this case, the coefficient of the x² term is 1, and the coefficient of the x term is 6. Therefore, the vertex of the quadratic is (-6/21, f(-6/21)) = (-3, f(-3)).
To find the value of f(-3), we can substitute -3 for x in the quadratic:
f(-3) = (-3)² + 6(-3) - 16 = 9 - 18 - 16 = -25
Therefore, the vertex of the quadratic is (-3, -25).