Evaluate the discriminant for each equation. Then use it to de...

Question

Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) See Example 9.

$9 x^{2} + 11 x + 4 = 0$

Accepted Answer

Hey, everyone in this problem, we are asked to identify the number of distinct solutions of the following quadratic equation by calculating its discriminate. And then to classify the solutions as either non real complex numbers, rationals or irrational. And the equation were given as X squared minus 36 X plus 324 is equal to zero. Our answer choices are a, the number of distinct solutions is one. The type of solutions is rational. B the number of distinct solutions is to the type of solutions is rational. See the number of distinct solutions is one and the type of solutions are irrational and D the number of distinct solutions is two and the type of solutions are non real complex. Now, let's go ahead and write out our equation. We want to think about what is the discriminate? Were asked to calculate the discriminate? What is that? Well, recall that if we have a quadratic function A X squared plus BX plus C is equal to zero, we can write the solutions to this equation as X is equal to negative B plus or minus the square root A B squared minus four A C All over two a. Okay. That's our quadratic formula. And you'll notice that the B squared minus four A C term that's under the square root of drawn in blue. And that's because that is our discriminate. Okay. So our discrimination will call it D is going to be B squared minus four A C. And it has this connection with the quadratic formula which gives us the solutions to our equation, which is why the discriminate can tell us the number of solutions and the type of solution. Alright. So let's start by figuring out what A B and C are for our equation. So the A is going to be the coefficient on the X squared term. So in this case, A is equal to one Okay. B is the coefficient on the x term. So B is going to be equal to negative 36 and don't miss those negative signs if they're there that can really change the number or type of solution you get if you get a sign error And C is equal to 320 for that constant term. Okay. So we have our A B and C values. Let's go ahead and calculate our discriminate. So D is equal to B squared minus four A C. This is going to be equal to negative 36 squared minus four times one times 324. This is gonna be 1, -1,296 and we get a discriminative equal to zero. Alright, so we know our discriminate but what does that mean? Well, when you're looking at the discriminate, you don't have to memorize all of these different cases if you remember that it comes from this quadratic formula. Okay. So we can look back and figure out if D is equal to zero, what happens to the solutions? Alright. Well, the discrimination zero, we take the square root of zero and we just get zero. Okay? And so our solutions become negative B plus or minus 0/2 A. And so the solution is just become negative B over two A. That means we only have one solution. Okay? Because we don't end up with two routes from our square root, we only end up with one route, which means we only end up with one solution. Okay. So when D A zero, we only have one distinct solution. Now we need to figure out if this is rational or irrational. Our one distinct solution recall is going to be a repeated root. That's what we often call it. And is it gonna be rational or irrational while we call that a rational number is one that can be written as an integer divided by an integer. So if our discriminate zero, the square root term goes to zero, we just get negative B divided by two A, that's an integer divided by an integer. And so we do indeed have a rational number. So we have one distinct solution and it is going to be rational. Alright. So if we look at our answer choices, this is going to correspond with answer choice. A the number of distinct solutions is one in the type of solutions rational. Thanks everyone for watching. I hope this video helped see you in the next one.

Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) x² - 8x + 16 = 0

Key Concepts

Discriminant of a Quadratic Equation

Number and Nature of Solutions Based on the Discriminant

Rational vs. Irrational Solutions

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