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.

$8 x^{2} - 72 = 0$

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 to classify the solutions as non real complex rational or irrational. The equation we're given is 13 X squared plus 44 X plus 15 is equal to zero. And the possible answer choices are a, the number of distinct solutions is to and the type of solutions is rational. B, the number of distinct solutions is one and the type of solutions are rational. See the number of distinct solutions is to and the type of solutions is irrational or D, the number of distinct solutions is two and the type of solutions are non real complex. Now, let's start with our equation 13 X squared plus X plus 15 is equal to zero. And let's recall what the discriminate is. We're asked to calculate the discriminate. What is that? Well, recall that if we have the equation of a quadratic function K in general A X squared plus BX plus C is equal to zero, we can write the solution to this equation. Okay as X equals negative B plus or minus the square root B squared minus four A C All over two a. And you'll notice that I wrote B squared minus four A C K that value under the square root in blue. And I've done that because that is the discriminate. Okay. So the discrimination is going to be equal to B squared minus four A C. And because it's part of this quadratic formula that gives you the solutions to the equation. It gives us information about the number of solutions and the type of solutions. Alright, so we have our equation. The first thing we need to do is figure out what our A B and C value are. So A is gonna be the coefficient with the X squared term. So A is equal to 13. In this case, B is going to be the coefficient with the X term B is equal to 44. Okay. And C is gonna be the constant term which is going to be equal to 15 here. Now substituting these values into our discriminate equation, we have the discriminate is equal to 44 squared -4 times 13 times 15 KB 2 -4 AC. This gives us -780 and we get a discriminative of 1156. Now this discriminate is positive, it is greater than zero. So what does that mean? Well, if we look at our quadratic formula, the discriminates under the square root. So if we have a positive value, then when we calculate our solutions, we're gonna end up with the square root of a positive number. When we take the square root of a positive number, we know that we're going to get to distinct roots. Okay. And that they're gonna be real. So this tells us that our solutions will have two of them and they're going to be real. So D is greater than zero, which tells us that we have two distinct solutions and they are real. All right. So now we need to figure out are they rational or are they irrational? And recall that a rational number is one that can be written as a fraction divided by or sorry, a number, an integer divided by an integer. Okay, we can write a rational number as a fraction of two integers. All right. So our description is 1,156. In order to calculate the solutions, we're going to take the square root of it. Okay. So let's look at the square root of 1,156. Now this works out to be 34. Okay. This square root is a nice integer value. So in our equation for X, we're gonna have some integer plus or minus some integer divided by an integer. Okay, by definition, that's a rational number. Okay. So because the square root the discriminate works out to be an integer. We are going to have rational solutions to our equation. Alright, so we found two distinct real solutions that are rational. If we look at our answer choices, that's going to correspond with answer choice. A the number of distinct solutions is to and 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.) 3x² + 5x + 2 = 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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