The Discriminant
Adding Subtractingand Multiplying Complex Numbers - Example 1
Complexand Imaginary Numbers - Example 1
Dividing Complex Numbers - Example 1
Findteh Minimumor Maximum Values - Example 1
Liberty University
Solving Quadratic Equationsby Completingthe Square


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quadratic equations by completing the square. Now we've got three steps in order to do this and the stuff how we're gonna do this is the first step is we're gonna Whatever the B term is, we're gonna find half of that. Be term. Step two is we're going to whatever we get in step one half of B, we're going to square that and then we're gonna add Step 22 X squared plus BX. So, basically, what that means is that we have X Square plus BX plus B over X squared would equal X plus B squared be half half a B squared. So it's always easier to understand a math problem when we actually get into with numbers. So let's look at our first problem X squared plus 12. It's so for this problem, we can make sure that equals zero. First step says we're gonna find half of bay. Well, RB term is 12, so half of 12 is six. So that's step one. Step two is we're going to square our solution. So our solution from step one with six and we're going to square that and that will be 36 and then Step three says we're gonna add 36 to X square plus 12 x So we'll have X squared plus 12 x, and we're gonna add 36 to that. Now, Once I do that, I can find the square root of my first term and my last term in order to find the factors and be able to solve. So let's do that. Well, the square root of X squared is X, and our square root of 36 is six. And I can tell you that 12 divided by two is positive. Six. So that means we're gonna use positive six as a factor. So we'd have X plus six squared would be are factors which means that if I went X plus six equals zero, I don't have to do it once because it would be the same. Both tops and subtract. Six X would equal negative six. So the solution to the equation would be negative. Six. So let's look at this one. We have x square plus eight x minus 20. Well, step a is we're gonna take our be term or be term is eight in half of eight is four. So first up A. We've got four from there. We're going to square four, and that would give us 16. Now, before we go any further, let's come back up to our equation because we need our equation. Toe actually, only have the two terms the A and B on that side. Well, it's got three terms. It's gotta C turn, eh? So what we need to do is we need to add 20 to both sides. So I have X Square plus eight x equals 20. So I'm going to add 16 to both sides. So we would have, because whatever we do to one side, we have to do to both. So we're going to have for steps. See, it would be X squared plus eight X plus 16 will equal 20 plus 16. So when I factor that out and so eight divided by two is positive for so the X Plus four squared would equal 30 six. From there, we can actually find our solution to both sides. So we would set this up a little differently because what we're saying is that X plus four squared with equal 36 so we need to find the square root of X Plus four squared and the square root of 36. Well, the square root of X Plus four squared is X plus four and square. Her to 36 is plus or minus six. So let's work that out. We're gonna work it out both ways. We're gonna work it out. X Plus four equal six and X plus four. What equal? Negative. Six. So it was attract four on both sides, so X could be, too. And X would be negative. 10. So our solutions or negative? 10. Positive, too. Let's work. Another example. X squared minus 10 X plus 24. Well, let's get it to where we only have the two terms that a term in the B term on the side. So we're gonna go ahead and subtract 24 on both sides. So that's gonna give me X squared minus 10. X equals negative 24. All right, for a first step or be term is negative. 10 and half of that is negative. Five for be term negative. Five squared is positive. 25. So for C, we're gonna add that to both sides so we'll have X squared. Modest 10 X plus 25 will equal negative. 24 plus 25. So now let's simplify that. So we have X squared. Modest 10 X plus 25 equals negative. 24 plus 25. Well, we can find the square, so we have X squared is gonna be eggs. Square root of five is going to be a plus or minus. I'm sorry. Square 25 plus or minus five. And negative 10. Divided by two is negative. Five. So we're gonna have X minus five. Squared What? Equal one. So that means square root of X minus five. He is X minus five and squared one is one plus or minus so that when we're solving this, we're gonna solve it as X minus five equals positive one X minus five equals native one. So we'll add first time we'll add five to both sides and eggs will equal six, and then we'll add five again and X will equal for. So our solutions would be four and six

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