Expert: Sherry Wallin - 4/25/2011

Question
Solve the following nonlinear inequality:

-3x^2-2x<=2x+4

Express the answer in interval notation.

Answer
Reemal~

The way the question is written there is no solution in the real numbers, thus there are no intervals on the real number line that will satisfy the inequality. I will show you why:

first move everything to one side and 0 on the other:
-3x^2-2x - 2x - 4 <= 0
-3x^2 - 4x - 4 <= 0

Now you can make everything positive if you like, I always recommend that one does that when they try to factor any equation, however, you can use the discriminant to determine what kind and number of solutions you have in any quadratic. The standard form of a 2nd degree (quadratic) equation is ax^2+bx+c = 0 where a~=0 (else it would no longer be 2nd degree because 0*x^2 = 0 and you have no squared term)

the discriminant says to calculate
b^2-4ac and if:

b^2-4ac >= 0 you have two real unique (different) solutions to the quadratic equation or

b^2-4ac  = 0 you have one identical (same) solution, example (x-2)^2 = 0 => x-2 = 0 and x-2 =0 so x = 2,2 but they are the same or

b^2-4ac < 0 you have a pair (two) of complex conjugate solutions example x = 2+3i, 2-3i

so in this problem a =-3, b =-4, and c = -4 sp b^2-4ac = (-4)^2-4(-3)(-4) = 16 -48 = -32 < 0 => only a complex conjugate pair for solutions, i.e., there is no real solution.

The Fundamental Theorem of Algebra basically tells you that at most you can have the degree of the polynomial number of solutions so since this is a quadratic, 2nd degree, at most, you will get 2 solutions, so you needn't look further and complex solutions always come in pairs because they are conjugates of one another so if you have one complex solution you really have two complex solutions and if a solution is complex there is no real number to be found that will solve the equation or to be found on the real number line which is what you are doing when you write out the interval for your solution.

Math Prof