Solve each equation in Exercises 66–67 by completing the square. ...

Question

Solve each equation in Exercises 66–67 by completing the square. 3x^2 -12x+11= 0

Accepted Answer

Hello today we're going to be solving this equation by using the method of completing the square. Now before we jump into this let's just go ahead and recall what completing the square is completing the square and completing the square states that if we have a mano me away in the form of a X squared plus bx we can get a term C which is equal to B divided by two squared and we can add that to the end of our mono meal to turn it into a factory trivial meal plus C. So this is the method we're gonna be using to solve this equation. So let's look at the equation that's currently given to us. The equation given to us is two X squared plus eight X plus seven equal to zero. Now in order to go ahead and start using the method of completing the square, we need to get a mono meal on the left hand side of this equation in order to do that, I'm just gonna go ahead and subtract seven to both sides of this equation. Doing so is gonna give me two X squared plus eight X. Is equal to negative seven and if you notice we have two and eight so we can go ahead and factor out a two from the left hand side. Doing so is going to give me two times X squared plus four X equal to negative seven. And just to get this mono meal on its own, I'm gonna divide both sides by two. Doing so is going to leave me with X squared plus four X equal to negative 7/2. Okay, now let's go ahead and start using the completing the square method in order to do that, we need to go ahead and identify what our B term is. We'll be is just gonna be the term in front of our X. So B is equal to four in this case and just to continue it over here, we want to go ahead and add a C to this mono meal. So C is going to equal to four divided by two. That value squared four divided by two is just two and two squared is going to give us four. So we're going to add four to both sides of this equation. The reason why I say both sides is because if we introduce 4 to 1 side of the equation, we need to balance it out by introducing four to the other side. So what we have now is X squared plus four, X plus four equal to negative 7/2 plus four. Now X squared plus four, X plus four is factory ble factoring, this is going to give me x plus two squared and we're gonna go ahead and add negative 7/2 and four together. Doing so is going to give me positive one half. Now, since we're still trying to solve for X, we need to go ahead and open up this pair of parentheses in order to do that, we're going to take the square root of both sides of this equation. Doing so is going to give me X plus two equal to plus or minus because keep in mind as square root produces two values plus or minus one over the square root of two. Now this is okay, but the only problem with this is that we don't want to leave a square root in the denominator. So we're gonna go ahead and rationalize this fraction in order to rationalize one over the square root of two, we're going to multiply the numerator and denominator by the square root of two. Doing so is gonna give me one times the square root of two, which is just the square root of two, All that over the square root of two squared and the square root of two squared is just gonna leave me with two. So we're left with the square root of 2/2. So that means that X plus two is going to equal to plus or minus the square root of 2/2. And in order to solve for X, I'm just gonna go ahead and subtract this two to both sides of this equation. Doing so is going to give me X is equal to negative two plus or minus the square root of 2/2. Now let's go ahead and combine both of these values together. If I put negative 2/1. What I want to do now is go ahead and find a common denominator between these two fractions. Well, the lowest common denominator here is going to be two. And in order to get that on the left hand side, I'm just going to multiply this fraction by two in both the numerator and denominator doing so is going to give me X is equal to negative 4/2 plus or minus the square root of 2/2. And combining both of these together is going to give me X equal to negative four plus or minus the square root of two. All over two. And that is going to be our solution to this equation. That also tells us that the solution to this problem is going to be be so I hope this video helps you in understanding how to use the method of completing the square and I'll see you all in the next video.

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