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Find the range of values for which x^2 +2x is greater than or equal to 8.

Probably the best thing to do is first plot the function, y = x^2 +2x, to get an idea of what you are dealing with. You'll see that it is bowl-shaped with one minimum at x = -1.

Clearly, the function grows larger as x moves away from -1 (both in the positive and negative directions) and will exceed 8 at two values of x. To find these values, it makes sense to find out for which values of x the function y equals 8. Another way to say this is to find the "roots" of the equation obtained by subtracting the number 8 from both sides of the equation for y above,

f(x) = y - 8 =

x^2 +2x - 8 = 0.

It is pretty easy to see that this equation "factors" into f(x) = (x+4)(x-2) = 0, from which it can be seen immediately that x = -4 and x = 2 solve this equation, that is, if you plug in either x=-4 or x= 2, then f(x) = 0. These values of x are called the roots of f(x). (You could have also used the quadratic equation to obtain these values).

So now we know the values of x for which y = 8. From the plot of the function, it is easy to see that if x > 2 or x < -4, then y > 8, which is the answer.

Randy

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