Answer: To turn the expression -x^2 - 4x - 4 into an always true inequality, we need to find its vertex. The vertex is the point where the parabola changes direction and is given by x = -b/(2a) and y = f(-b/(2a)), where f(x) is the function in standard form, ax^2 + bx + c.
In this case, a = -1, b = -4, and c = -4, so the x-coordinate of the vertex is:
x = -(-4)/(2*(-1)) = 2
To determine whether the vertex is a maximum or minimum, we can look at the coefficient of x^2. Since a = -1, the parabola opens downward and the vertex is a maximum.
Therefore, we can write the inequality:
x^2 - 4x - 4 > -2
This is because the y-coordinate of the vertex is f(2) = -2, and the expression -x^2 - 4x - 4 is always less than -2 when x is not equal to 2. When x = 2, both sides of the inequality are equal.
So the inequality -x^2 - 4x - 4 > -2 is true for any value of x.
Explanation: