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. 9x^2 + 11x + 4 = 0

Accepted Answer

Hey, everyone. Welcome back in this problem. We are 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 into either non real complex numbers, rationals or irrational. The equation we're given is 12 X squared minus five X plus two is equal to zero. And our answer choices are a, the number of distinct solutions is to and the type of solutions are rational. B the number of distinct solutions is two and the type of solutions is irrational. See the number of distinct solutions is to the type of solutions not real complex. And D the number of distinct solutions is one and the type of solutions are rational. Alright, so we have our equation 12 X squared minus five X plus two is equal to zero. Now, what is the discriminate? Well, recall that if we have A X squared plus BX plus C is equal to zero. Okay. The equation for general quadratic and we want to find the solutions, we can do that using the quadratic formula and we get that X is equal to negative B plus or minus the square root of B squared minus four A C over to A and you can see that the B squared minus four A C that's under that square root of drawn in blue. And that's because that is our discriminate. Okay. So we're discriminated, we're gonna call D is going to be equal to B squared minus four A C. Okay. And because it's in this equation for X and A quadratic formula, you can see how it's going to relate to what type of solution we have. All right. So looking at our equation, let's first figure out what A B and C R A is going to be the coefficient on the X square term. So in our case A is 12 B is the coefficient on the X term. Now be careful here, we have negative five X and so B is going to be negative five can make a big difference if you miss those signs on what type of solution you end up with. And see is that constant term? And that is going to be too okay. So we have A is equal to 12 B is equal to negative five and C is equal to two. Now, we can substitute those numbers into our formula for the discriminative. We get that the discriminative is equal to B squared minus four A C which is going to be negative five squared minus four times 12 times two. This is going to be equal to -96, which gives us a discriminate of -71. Now what happens if we have a discriminate that's negative and you don't have to memorize every single case with the discriminate. Remember that it comes from this quadratic formula equation for the solutions. If we look back at this quadratic formula equation, we found that D is negative. So this value underneath the square root is going to be negative. What happens when we take the square root of a negative number? Well, we know that we're gonna get two distinct solutions but they're going to be complex. When we take the square root of a negative number, we get complex roots and they come in Paris. Okay. And so this means because D is less than zero, we're gonna have two distinct solutions and they are going to be non real complex. Okay? Alright. So if you can't remember the equation or the formula for the discriminate, you can always just go back and look at your quadratic formula and get it from underneath the square root. And you can use that information to guide what the solutions look like. You don't have to memorize a ton of different cases. So if we look at our answer traces, we have answer choice. E the number of distinct solutions is two and the type of solutions is non real complex. Thanks everyone for watching. I hope this video helped see you in the next one.

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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. 9x^2 + 11x + 4 = 0