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

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. 8x^2 - 72 = 0

Accepted Answer

Hey, everyone in this video, we're asked to identify the number of distinct solutions of the following quadratic equation by calculating its discriminate. Then we're asked to classify the solutions non real complex number, irrational or irrational. And the equation we're given is three X squared minus 588 is equal to zero and are possible answer choices. We have choice. A, we have that the number of distinct solutions is to and that the type of solutions is rational. We have B the number of distinct solutions is to and the type of solutions is irrational. We have seen the number of distinct solutions is one and the type of solutions is rational and we have d the number of distinct solutions is to in the type of solutions of non real complex. Alright, so we have our quadratic three X squared minus 588 is equal to zero. Now, what is the discriminate? Okay. We're asked to calculate the discriminate. What is the discriminate? Well, recall if we write our quadratic as X squared plus BX plus C is equal to zero in general, the quadratic formula that finds the solutions for X is going to be X is equal to negative B plus or minus the square root of B squared minus four A C all over to A. Okay. That's the quadratic formula to find the solutions for X. And the discriminative D is given just by what is under the square root. So the discriminative we're gonna write it as D is just B squared minus four A C. Okay. So looking at the equation we're given, we have three X squared minus 588 is equal to zero. Okay. Comparing this to the general equation, this tells us that A is going to be equal to three B is going to be equal to zero. Okay? We don't have an X term, we have an X squared term and a constant term and C is going to be equal to negative 588 and don't miss those negative signs if they're in your equation. Okay. They need to go with that coefficient. So we have our A B and C. Let's go ahead and calculate our discriminate. So D is going to be equal to B squared minus four A C which is equal to zero squared minus four times three times negative 588. Okay. So the first term is going to be zero and we have negative four times three times negative 588. This is going to give us a discriminate of 7,056. So our discriminate is positive. Okay. So the discriminate is greater than zero, it's positive. What does that mean about the roots? Well, let's look back at our quadratic formula. Okay, the discriminate is under the square root. So now we have a value that's positive. We want to take the square root. We know if we take the square root of a positive value, we're going to get two distinct real numbers. Okay. So what that tells us is that we are going to have two distinct real solutions. Now, our answer choices also distinguished between having a rational solution and an irrational solution. So let's think about that recall that a rational number is one that you can write as an integer over another integral. So if we look at our quadratic formula, we can see that if the square root works out to an integer, then we just have an integer plus or minus an integer divided by an integer. That is by definition a rational number. Okay. If the square root doesn't work out to a nice integer, then we need to leave it as the square root that's going to be an irrational number. Okay. So let's look at the square root of our discriminate. Okay. The square root Of is actually equal to 84. This is an integer, which means that when you're finding your solutions, you just have an integer divided by an integer. Okay. And so the solutions will be rational. And so what we found is that we have two distinct real solutions and they are rational. We go back up to our answer choices. We see that that corresponds with answer choice, a, a number of distinct solutions to type of solutions. Rational. Thanks everyone for watching. I hope this video helped see you in the next one.

Struggling with College Algebra?

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. 8x^2 - 72 = 0

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