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

Question

Solve each equation in Exercises 47–64 by completing the square.x^2 - 5x + 6 = 0

Accepted Answer

Welcome back. I'm so glad you're here. We've got this equation, X squared minus 16 X plus 63 equals zero. We're asked to solve it by completing the square. Well, notice that our equation is in the format A X squared plus bx plus C equals zero. Where are a term in front of the X squared is one. B term is negative 16 and R C term is 63. Well, the first step in completing the square is we want to ask ourselves is are a term equal to one. And yes, we just said that it is and that makes this a lot simpler. So the next step when are a term is equal to one is that we're going to scooch that C term over to the right hand side of the equation. So in this case we're going to take 63 and subtract it from both sides. So this will give us X squared minus 16 X. And then we're going to leave a bunch of space equals negative 63. And for our red equation here that would look like X squared plus B X. Leave a bunch of space equals a negative C. So how do we fill in that space? Well, what we're going to do is we're going to take our B term, We're going to divide it by two and we're going to square it and again it's only be divided by two squared because our a term is equal to one. If the a term was not equal to one, it would be a little bit more complex. So in this case our B term is negative 16 And we're going to divide that by two and square it And -16 divided by two is -8. And before we do the squaring step, I want you to take note of what the number is inside the parentheses and just hold onto it for a moment and then stop holding it because we're going to continue with the squaring negative eight squared is 64 that is the term that is going to get filled into that blank. So let's do that. X squared minus 16. X plus 64 equals negative 63 plus 64. And now we take a look at that X squared minus 16. X plus 64. That's pretty easily factory ble. And on the right hand side, that's just going to equal one. Negative 63 plus 64. So on the left, what times what gives us X squared. That's X times X. What times what equals a positive 64 when multiplied but adds up to a negative 16. That's minus eight minus eight. Or in other words x minus eight times x minus eight is x minus eight squared. Look, we have completed that perfect square. And yes, this was by design because you might notice that this number in here in the parentheses, is that term that we highlighted earlier. This is because by definition when you've got X squared plus b, X plus b over two squared. Let's rewrite that. There we go. That is the same as parentheses. X plus B over to end parentheses squared. There's our highlighted number and that's exactly what it looks like. Wonderful. So the next step here is just to get X by itself, we can take the square root of both sides to get rid of that squared. So the square root of x minus eight squared is x minus eight equals and then we take a plus or minus because we took the squared of one in the squared of one is just one. And now note that we've got two equations here. We are going to solve both by adding eight to both sides. And because the plus or minus, we have an X equals a positive one plus eight and we have an X equals a negative one plus eight. Well, one plus eight is X equals nine and negative one plus eight is X equals seven. We look at our answer choices and this matches with answer choice D. Well done, we'll catch you on the next one.

Solve each equation in Exercises 47–64 by completing the square.x^2 - 5x + 6 = 0

Key Concepts

Completing the Square

Quadratic Equations

Discriminant

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