Solve each equation in Exercises 47–64 by completing the squar...

Question

Solve each equation in Exercises 47–64 by completing the square. $x^{2} + 3 x - 1 = 0$

Accepted Answer

Welcome back. I'm so glad you're here. We've got this equation, we have X squared minus 11, X plus one equals zero. And we're to solve it by completing the square? I'm going to go through all the steps of completing a square. But if I'm as I'm going through this, it's a little bit fuzzy, definitely go back and check some previous lesson videos on completing the square for a little bit of a refresher but otherwise continuing on um I want to point out that this is in the format of a X squared plus bx plus C equals zero. So I'm going to be referring to the A terms, the B terms in the sea terms which are so A in this case is one, B is negative 11 and C is also one. The first step in completing the square is we want to ask ourselves does a equal one and in this case, yes, it does. That makes it a lot simpler. So for our next step we don't have to get eight equal one. It already does. We're going to take that C term and we're going to scoot it over to the right hand side of the equation. So in this case we're going to subtract one from both sides. So now we'll have X squared minus 11 X. And where that C term was, we're going to leave a blank space and it equals negative one. So it's very important to leave that space there because the next step of completing the square is going to be to fill in that blank space. So what are we going to fill in that blank space with, What we're going to do is we're going to take our B term and we're going to divide it by two and we're going to square it. And just pointing out that if you're a term wasn't one, this would be a little bit more complex. But since A is equal to one, we're going to take the B term divided by two and square it. So what is our B term? Our B term is -11. We're going to divide it by two and we're going to square it and whenever you have that be term inside the parentheses and you haven't squared it yet. But it's simplified in the parentheses. You want to pause for a moment and take stock of what that number is and then you can cease pausing and continue on. We're going to go ahead and square it. So negative 11 halves squared equals 121 4th. And this is the number that is going to get filled in to the blanks. So let's go ahead and write that. We have X squared minus 11, X plus 121 4th equals negative one plus 121. Force. Very important to add that to both sides of the equation. So our red formula here is looking like X squared plus bx plus B over two squared equals negative C plus B over two squared. And now we want to recall from previous lessons that by its very definition when we're completing the square, when we have this X squared plus bx plus B over two squared. That's what we've got here with the X squared minus 11 X plus 121 4th. That's the same thing as parentheses. X plus the number we highlighted B over two end parentheses squared. This is where that number you had to remember is so let's rewrite that over here instead of X squared minus 11 X plus 121. Force its parentheses X plus are highlighted number negative 11 halves and parentheses squared equals and we can simplify on the right hand side negative one plus 121 4th is 117 4th. Cool. Now this looks like something we can solve for X. We can take the square root of both sides and the square root of X plus negative 11 halves squared is just X plus negative 11 halves. The square root and squared cancel out equals well the square root of 117 is going to simplify to three square root 13 but actually let me leave a little bit more space there because when we take the square root it's going to be plus or minus. So the numerator is three square root 13 and the denominator the square root of four is two. So now to get the X by itself, we're just going to add 11/2 to both sides. And we're dealing with two equations because of that plus or minus. So one of the equations is X equals positive three square to 13/ plus 11/2. And the other equation is going to be X equals the negative three squared of 13/2 plus 11/2. And because they have a common denominator you can combine those. Um The numerator are not like terms. So it'll just be three squared of 13 plus 11. All divided by two. And over here you'll just have negative three squared of 13 plus 11, all divided by two. We look at our answer choices and this matches with answer choice B. Well done we'll catch you on the next one.

Solve each equation in Exercises 47–64 by completing the square. $x^{2} + 3 x - 1 = 0$

Key Concepts

Completing the Square

Quadratic Equations

Isolating the Variable

Watch next

Solve each equation in Exercises 47–64 by completing the square. x2+3x−1=0x^2 + 3x - 1 = 0

Key Concepts

Completing the Square

Quadratic Equations

Isolating the Variable

Watch next

Solve each equation in Exercises 47–64 by completing the square. $x^{2} + 3 x - 1 = 0$