Solve each equation using completing the square. See Examples 3 a...

Question

Solve each equation using completing the square. See Examples 3 and 4. -2x^2 + 4x + 3 = 0

Accepted Answer

Welcome back. I am so glad you're here. We are given this equation negative two X squared plus 26 X minus 81 equals zero. And we are asked to complete the square and then solve it well for completing the square first. We want to see what format our equation is in and it is in the format of a quadratic equation where we have a X squared plus bx plus C equal to zero. And now to complete the square we want to ensure that our a term what's in front of the X squared is equal to one. Here are a term. The one in front of the X squared is negative two. We need that to be a one. How do we do that? We're going to divide it and every other term here by negative two. So we'll have negative two X squared divided by negative two which equals one X squared or just X squared. Great. Now we have plus 26 X divided by negative two. That's going to be a minus 13 X. -81, divided by -2. That will be a plus. And we'll keep it as 81/2 and it's still equal to zero because zero divided by negative two still equals zero for our next step. We want to rearrange this so that we bring our C term over to the right hand side of the equation. And when we do that we want to leave a space where are C term was and a space after the opposite signed C term on the right hand side as well. So let's bring our negative or we'll bring our 81 over to by subtracting 81/2 from both sides. And then that will give us X squared minus 13 X. And then leave a space equals a negative 81/2. And then leave a space. The next step in completing the square is to fill in that space. We will fill in that space by taking our B term. In this case B is negative 13, that's always in front of the X. To the first power. So we have negative 13. We're going to divide it by two and then we are going to square it. And if we take negative 13 squared that equals a positive 169 and two squared equals four. Now we are going to add that, add 169 over four into both of these spaces that we had left open. And next we are going to factor the left hand side of the equation but we can do it pretty quickly because our terms inside here are going to be X for the first term and for the second term it's going to be this be over to inside the parentheses. So it's going to be negative 13/2. So X plus negative 13/2. And that is squared. If we were to foil that out it would equal this X squared minus 13 X plus 169 over four all by design so that that was completing the square. Now, we just need to solve this. So we need to add the fractions on the right hand side of the equation. They need to have a common denominator. In this case it will be four. So we can multiply the first fraction, the negative 81/2 by 2/2 in the numerator negative 81 times two equals negative 162. And in the denominator two times two equals four giving us our common denominator. And now we can add the two fractions on the right hand side. Going to bring down the left hand side X. And I'm going to put minus 13/ squared equals because we have a common denominator on the right hand side, we can add our numerator negative 162 plus 169 equal seven. Still over four. And now to solve for X, we need to get X by itself, we can take the square root of both sides of the equation to get rid of that squared on the left. So now it will be x minus 13/ equals because we take the square root on the right hand side, we have to have plus or minus and we can distribute that square root to both the numerator and the denominator. So it will be the square root of seven over the square root of four in the denominator is two And we're getting close, X is almost by itself. Now we'll add 13/2 to both sides and X is by itself on the left hand side. Yea, X is so happy and that equals, I'm going to put the 13 over to first plus or minus the square root of 7/2. So we have two different answers here. One is going to be where X equals 13/2 plus the square root of 7/2. And the other is X equals 13/ minus the square root of 7/2. Because we have a common denominator for both of these answers, we can combine our fractions, we can take the 13 plus the square root of seven in the numerator for the first one and that's all over two. And we can take the 13 minus the square root of seven in the second one and that is all over two. And there is our answer, We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one

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Solve each equation using completing the square. See Examples 3 and 4. -2x^2 + 4x + 3 = 0

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