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

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

x^2 + 4x + 1 = 0

Accepted Answer

Welcome back. I'm so glad you're here. We have the equation X squared minus two. X minus 23 equals zero. And we are to solve this by completing the square. While I want to note that we have an equation in the format A X squared plus B X plus C equals zero. Where A corresponds to one in front of the X squared B corresponds to negative two in front of the X and C. Our constant term is negative 23. And the first question we ask when completing the square is does a equal one and in this case yes, it does. Which makes this much quicker of a process than if they were not equal to one. So because A equals one, the next step will take for completing the square is to scoot R C term over to the right hand side of the equation. So C is negative 23. We're going to add 23 to both sides and then our equation will read x squared minus two X. And then we're going to leave a space where that C term was equals 23. And we're going to leave a space after that C term on the other side too. So for our red here, this will look like X squared because that A equals one plus B. X. Space equals negative C. And for the next step we're going to fill in that space, what do we fill it in with? Well we're going to take our B term. We're going to divide it by two and we are going to square it. So what is our B term? Our B term is negative two. We're going to divide that by two and we're going to square it. What's negative two divided by two. That's negative one squared. And before you do the squaring step, I want you to always take note of the number inside the parentheses once it's simplified, but before it's squared and once you've taken note you can continue on because we do need to go through and square it. So negative one squared. That is a positive one. And that is the number that is going to go in our space. So let's fill that in X squared minus two. X plus one equals 23 plus one. Well look at that X squared minus two X plus one. We can factor that. And on the right hand side of the equation, 23 plus one is 24. So what times what gives us X squared. That's x times X. And what? Two factors when multiplied, equal a positive one. And when added equal and negative two. That's a minus one minus one. Or in other words X minus one times X minus one is x minus one squared. Look at that perfect square there. And also note this number inside the parentheses with our X. That's the same as the number that we took note of a little bit earlier. That is because when completing the square, when you have X squared plus bx plus B over two, a squared And note this is when a equals one, it is always going to be parentheses X plus. That highlighted term be over to end parentheses squared. Well now looking back at our problem, we can go ahead and solve for X. We've got x minus one squared. We can take the square root of that and then take the square to the other side as well. So on the left, the squared of x minus one squared is just x minus one and that equals well we took the square root of 24 so now it's plus or minus and the square root of 24 is two square root six. Now we want to add one to both sides and note that we have two equations because X equals a positive two squared of six plus one and X also equals a negative two squared to six plus one. Those aren't like terms, you can't combine them so that's as simplified as it's going to get. But we can rearrange the numbers on the right hand side so that it looks more like the answers. So we'll have X equals a one plus two squared of six and we have an X equals one minus two squared of six. We look at our answer choices and that matches with answer choice. A well done. We'll catch you on the next one

Solve each equation in Exercises 47–64 by completing the square. x^2 + 4x + 1 = 0

Key Concepts

Completing the Square

Quadratic Equations

Perfect Square Trinomial

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