Which is smaller x^2+4 or 4x-4

To determine which expression is smaller, we can set them equal to each other and solve for x.

x^2 + 4 = 4x - 4

To simplify, let's subtract 4x from both sides and add 4 to both sides:

x^2 - 4x + 4 = 0

We can further simplify by factoring the left side of the equation:

(x - 2)(x - 2) = 0

Setting each factor equal to zero:

x - 2 = 0

x = 2

So, when x = 2, both expressions are equal.

Now, we can examine the behavior of the expressions before and after x = 2.

Let's consider values of x less than 2:

For x < 2, the term x^2 + 4 will be larger than 4x - 4, because x^2 will be positive while 4x will be negative.

Let's consider values of x greater than 2:

For x > 2, the term 4x - 4 will be larger than x^2 + 4, because 4x will be greater than x^2.

Since both expressions are equal at x = 2, we can conclude that 4x - 4 is smaller than x^2 + 4 for x ≠ 2.