Solving Quadratic Inequalities

Video Transcript

"Hi, my name is Samir Malik. I'm a tutor for middle school and high school students in the Austin, Texas area. Today I'm going to describe for you how to solve quadratic inequalities. Quadratic inequalities are used commonly in algebra to trigonometry classes, as well as in pre-calculus classes, as well on a high school level. So it's definitely very a crucial element, and it would help to know how to solve these quadratic inequalities, especially in these types of courses. I like to illustrate this in an example, so you can actually see how it is done once I write down a sample equation for you. For quadratic inequalities, we can do AX square, plus BX, plus C is less than zero. And we can do AX square plus BX plus C is less than or equal to zero. We can do AX square plus BX plus C, which is greater than zero. And we can do AX square plus BX plus C, which is greater than or equal to zero. These are different types of quadratic inequalities that we can do to illustrate it. Now I'd like to throw in some numbers for you so you get an example of how it all works. So we can do, for example, 2X square plus 4X plus 1 has to be less than 0. So let's go ahead and solve for X, here, by using the quadratic formula which is X equals negative 4, plus or minus, 4 square minus 4 times 1 times 2, which equals negative 4. This is all over 2 times 2, which equals negative 4 plus or minus the square root of 16 minus A, which is the square root of A, all over 4. So we have, and the negative 4 and this 4 divide out to negative 1. So we have negative 1, plus or minus, square root of 8 over 4, so we're able to know that X equals negative 1 plus square root 8, equals 1.828 divided by 4, which is 0.46, or we have negative 1 minus square root of 8, divided by 4, which equals negative 1. Now, we plug that into our equation, so we have 2, 0.46 squared, plus 4, 0.46 squared plus 4. 0.46 plus 1, which equals .46 squared times 2 equals .423 plus 4 times .46, which equals 1.84 plus 1, which equals 3.263. And then we have 2, negative 1 square plus 4 negative 1, plus 1, which equals...negative 1 square is 1, so we're 2 minus 4 plus 1, which equals negative 2 plus 1, which is negative 1. So we have our X values for this quadratic to be equal to 3.263 or negative 1. So this is how we solve for quadratic inequalities. Pretty much we need to take the equation that we have, the quadratic inequality, we need to take the quadratic formula and solve for that quadratic inequality by solving for X. Once we've solved for the value of X, all we do is we plug that X value back into our inequality to make the inequality true, and to check to see whether that value for X makes our inequality true."