Solve by completing the square: 2x² – 5x + 1 = 0.

Question

Accepted Answer

Welcome back. I'm so glad you're here. We have the equation three X squared minus eight X plus four equals zero. And we are asked to solve it by completing the square. Okay, when we are completing the square we typically are working with formulas in the quadratic equation. The format of a X squared plus bx plus C equals zero. And the first question that we ask ourselves is does a equal one? And in this case no A does not equal one A equals three. So what we're going to do when a does not equal one is we are going to divide every term by a. So on the left hand side here we're dividing every term by three. So we've got X squared minus eight thirds. X plus four thirds equals still zero. And on the right this is X squared because a divided by A is one plus B over a X plus C over A equals zero. And the next step is that we're going to scoop this C term to the right hand side of the equation will subtract C over A from both sides. And on the left hand side we're going to subtract four thirds from both sides. And then very important we're going to leave a space where that C term was So we've got X squared minus eight thirds. X. Space equals negative four thirds. And then a space after that C term on the right hand side as well. So for our red here that looks like eight or sorry excuse me, X squared plus B. Over a X. Space equals negative C over A. And now in these spaces we are going to fill them in. What are we going to fill them in with? We're going to take our B over A term. We're going to divide it by two and then we are going to square it. So on the left here, what's our B term? There be over A. That is negative eight thirds. We're going to divide that by two and we are going to square it. I'm going to rewrite this negative eight thirds divided by two this way, because that helps me remember that when we have two fractions and we're dividing them, that's the same as multiplying by the reciprocal of the second one. So it's negative eight thirds times one half squared. And that negative eight and the two reduce each other so the two becomes a one and the negative eight becomes a negative four. And we've got multiplying straight across because it's being multiplied by 1/1. We now have negative four thirds squared. And I want to pause right there. Whenever you have a simplified number inside the parentheses before you square it then take note of what that number is. Hold on to it for just a moment because we will need it in a minute. But for now we're going to go ahead and finish that squaring. So negative four squared is going to be a 16 in the numerator and three squared is going to be a nine in the denominator and we are going to take this number 16/9 and it is going to fill in that space. So now we will have X squared minus eight thirds X plus 16 9th equals negative four thirds plus 16 nights. We need to add that 16 9th to both sides because when we add it to both sides it's like we're not even changing it what we do to one side. We need to do to the other. Okay, now looking at this X squared minus eight thirds X plus 16 9th. We can factor that and on the right hand side negative four thirds plus 16 9th. That's going to be 4/9. So factoring the left hand side, what times what gives us X squared. That's going to be X times X. What? Two numbers when multiplied? Give us 16 9th a positive 16 9th. But when added together together, give us a negative eight thirds. Well that is going to be minus four thirds and minus four thirds. And we can rewrite that X minus four thirds times X minus four thirds as x minus four thirds squared. And that equals 4/9. And look at this beautiful perfect square that we have. And look on the left. Look at this number here inside our parentheses, that is the number that we took note of before. So this be over a over to that is the number that gets to go into our parentheses and helps form our perfect square. Now we can sell for X. I'm going to rewrite this up here and do that work up here. So we have x minus four thirds squared equals four. Nineths to get X by itself. First we're going to start by taking the square root of both sides. The squared of x minus four thirds squared is just x minus four thirds and that equals okay now we take plus or minus because we're taking the square root and the square root of four is two and the squirt of nine is three. So we have x minus four thirds equals plus or minus two thirds. That means we've got two equations. Now I'm going to add four thirds to both sides to get X by itself and then I'm going to write there are two equations from the plus or minus. We have X equals the positive two thirds plus four thirds and we have X equals the negative two thirds plus four thirds. So starting with the 1st 12 3rd plus four thirds is six thirds or two. So one value for x is two negative, two thirds plus four thirds is two thirds. So our two X values are X equals two and X equals two thirds. We look at our answer choices and that matches with answer choice. C well done. We'll catch you on the next one

Solve by completing the square: 2x² – 5x + 1 = 0.

Key Concepts

Completing the Square

Quadratic Equation

Discriminant

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