Solve each quadratic inequality. Give the solution set in interva...

Question

Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. 4x^2+3x+1≤0

Accepted Answer

Hello everyone. In this video, we're going to be looking at this practice problem which states for the following quadratic inequality solve and express the solution set in interval notation. And the quadratic inequality is nine X squared plus six X plus two is less than or equal to zero. And in this case, we're given four answer choices. You can see that answer choice A covers only negative numbers because it goes from negative infinity only up to zero. And B covers all numbers except for zero and C covers all real numbers. And then D is no numbers or no solution. So in order to figure out which of these is correct, let's go ahead and take a look at our inequality which is nine X squared plus six X plus two Less than or equal to zero. So in order to solve for X, I'm going to try and complete the square. But before I do that, I'm going to get rid of this coefficient here nine. So if I factor out a nine from this equation, I have my first term nine X squared divided by nine is simply X squared. If I divide nine from six, I have 6/9 X-plus 2/9 less than or equal to zero. And you can see I can simplify six divided by nine to be two thirds. So now I'm going to try and complete the square within this matrix. So I'm going to group X squared and two thirds X. So this will look like nine times X squared plus two thirds X. And remember when we're completing the square, we're adding a number here. So I'll leave the space there and I still have the 2/9 on the outside and the less than or equal to zero. So the number that I need to add here is denoted A squared. And well, where does a come from? It comes from this middle term here. So I have to a is equal to two thirds. So in order to solve for a, I'm gonna divide by two. So that means a is equal to 2/3 divided by two, which is 1/3. So the number that I need to be adding is one third squared, Which is 1 9th. So because I added the 1 9th in here, I can't just add numbers to my equation Because I added 1 9th inside the parentheses. I need to subtract 1/9 outside the parentheses. So if I write my inequality, it's now nine times X squared plus two thirds X plus 1/9. And now we have two X, 2/9 minus 1/9 is 1/9. We still have less than or equal to zero. So I'm going to go ahead and distribute this nine to first this equation here and then to the fraction out here. So my equation will become nine times parentheses, X squared plus two thirds, X plus 1/9 and parentheses plus nine times 1/9 less than or equal to zero. And you can see that nine times 1/9 is equal to one. So from here, I'm going to factor this equation here. So the reason that I completed the square was so that I can factor this equation and solve for X. So when we factor this equation, I want something that looks like this. So my first term is X squared. So I know that X times X gets me X squared. And the second term is actually a which I know is one third. So if I rewrite this inequality or this equation with my factored form, I'll have still the nine on the outside. But now I'm going to factor that equation. So I have X plus one third squared plus one is still less than or equal to zero. And now you can see that I have X without any power. So I'm going to start to solve for it. I'll start by subtracting one from both sides. So that leaves me with X or nine times X plus one third Squared less than or equal to negative one all divided by nine. So this leaves me with X plus one third squared, less than or equal to negative 1/9. And from here, you might want to take the square root to cancel out the squared here from both sides. But if we stop to look at the equation right now, you can see that we have a squared number here. So whenever you square a number or whenever you square negatives, you should get a positive. But in this case, we have a negative 1/9. So this is already a not true statement because whenever you have something squared, it's always going to be positive because it's the same number squared. So say I had X equals to negative 2/3 this parentheses would become negative one third. But because it's squared, it's going to give me a positive number. But you can see that I have negative 1/9 on the right. So this inequality is already not true, which means that the inequality has no solution. And if we look at the answer choices, you can see that D is also no solution. So hopefully this video helped and see you next time.

Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. 4x^2+3x+1≤0

Key Concepts

Quadratic Inequalities

Interval Notation

Factoring Quadratics

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