In Exercises 71–72, without solving the given quadratic equation,...

Question

In Exercises 71–72, without solving the given quadratic equation, determine the number and type of solutions. 9x^2 = 2-3x

Accepted Answer

Hello today we're going to determine what type of solutions we have from a quadratic equation without actually solving it. So the quadratic equation given to us is four X squared equal to five plus six X. Now in order to continue we just need to go ahead and set this equation equal to zero in order to do that. We're going to subtract two negative six X from both sides of the equation and we're gonna subtract five from both sides of the equation. Doing so is gonna leave me with four X squared minus six x minus five equal to zero. Now how do we determine what type of solutions we have without actually solving this. We're gonna use something called the discriminate and the discriminate comes directly from the quadratic equation. The discriminate is B squared minus four times a times C. So what we need to do now is go ahead and determine what are B. A and C are. Well A. Is just a coefficient in front of our X square term. So A is going to equal to four. B. Is our coefficient in front of our X term. So B is going to be equal to negative six and C. Is just the constant at the end of this. Try no meal and in this case C is equal to negative five. Now we just need to go ahead and take these valleys and plug it directly into the discriminate. Doing so is going to give me negative six squared minus four times four times negative five. Now let's go ahead and simplify negative six squared is going to be 36 four times is going to be 16 times negative five and 16 times negative five is going to give me negative 80. So now we have 36 -280 And we have minus negative number. So this is going to become a positive value. So we're left with 36 plus 80 and that's going to give us the value of 116. So what does this value tell us? Well, the conditions for the discrimination is that if the discriminate produces a value that is greater than zero, we have two solutions. If the discriminate it produces a value that is equal to zero. We have one solution. And if the discriminate produces a value that is less than zero. There are no solutions In this case. Here we are left with 116 which is clearly greater than zero. That means that we are going to have two solutions and that means that the solution to this problem is going to be D So I help this video helps you understand what the discriminate does for you. And I'll go ahead and see you all in the next video

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