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. 4x^2 = -6x + 3

Accepted Answer

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 into non real complex number, rational or irrational. Now, the equation we're given is seven X squared is equal to 11 X minus three. And the answer choices are a, the number of distinct solutions is to and the type of solutions is irrational. B the number of distinct solutions as to the type of solutions is rational. See the number of distinct solutions is one and the type of solutions is rational and D the number of distinct solutions is to and the type of solutions are non real complex. So first things first, we're gonna write out our equation seven X squared is equal to 11 X minus three. Now, what is the discriminate? Okay, let's recall if we have the quadratic A X squared plus BX plus C is equal to zero, we can write an equation for the solutions to this equation. OK. X equals what X equals negative B plus or minus the square root of B squared minus four A C over to A. Okay, that's your quadratic formula. Now, I've drawn B squared minus four A C okay. That value under the square root in blue because that is your discriminate. Okay. So the discriminate D is given by B squared minus four A C and that's what we're asked to calculate. So if we look at our equation, we want to find the value of A B and C first, but we have to be careful the A B and C They are given when they are all on one side. And this equation is equal to zero. Okay. In our case, we have the seven x squared on one side, we have 11 X -3 on the other side. So we need to first move some of these terms over. So let's move everything to the left hand side. We have seven X squared, we need to subtract 11 X to move it over to the left hand side. And then we need to add three to move the negative three over to the left hand side. And so we get seven X squared minus 11 X plus three is equal to zero. So the A is going to be the coefficient with the X squared term. So A is going to be equal to seven. In this case, B is going to be the coefficient with the X term. In this case, it's gonna be negative 11. Okay. And don't miss those signs. That's important. And it can make a big difference on the type of solutions you get in your discriminate And C is going to be three. Okay. So if we calculate our discriminate, now we have that the discriminate D is going to be equal to negative 11 squared minus four times seven times three. So it's gonna give us 121 -84. We get a discriminate of 37. Okay. So let's start with how many solutions do we have? Okay. Well, this is a positive value. The discriminates positive. So if we're calculating solutions, let's go back up to our quadratic formula. And in the equation for a quadratic formula, we're taking the square root of a positive number. If we take the square root of a positive number, we know that we are going to get two distinct solutions and they're going to be real. So D is 37, which is greater than zero, which tells us that we have to distinct real solutions. And that's really handy to know where the discriminate comes from from that quadratic equation formula. So that you don't have to memorize all of these different cases. You can look back at that formula and say what happens if my discriminate is positive? Okay. Well, you're taking the square root of a positive number. All right. Now we have to figure out if these are rational or irrational And recall that a rational number is one that you can write as an integer divided by an integer. Okay. So again, let's look at our quadratic formula. We're taking the square root of our discriminate here. So we need to take the square root of 37. Well, the square root of 37 does not work out to be a nice integer. Okay, we have to leave it as the square root of 37 unless we approximate it, which tells us that this is going to be an irrational value. Okay. So the square root of 37 is not an integer. So our solutions are irrational. All right. So we found that we have two distinct real solutions and that they're going to be irrational. And if we look back at our answer choices, we see that that's going to correspond with answer choice. A the number of distinct solutions is to and the type of solutions irrational. 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.) See Example 9. 4x^2 = -6x + 3

Key Concepts

Discriminant

Types of Solutions

Standard Form of a Quadratic Equation

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