![]() ![]() Four times 21, no, that difference between four and 21 is But the differenceīetween these two numbers, if one was positiveĪnd one was negative is a lot more than eight. Would have to be negative, one of them would have to be positive in order to get to negative 84. Look at the factorization of negative 84, of 84 generally. So they'd have to have different signs, since when I take their product And think about what two numbers if I multiply them I get negative 84. ![]() We could've said x squared minus eight x minus 84 is equal to zero. Now another way that weĬould've approached it without completing the square. To both sides of this and you get exactly this This part right over here is minus 100 is equal to zero. This part right over here, this is x minus four squared. Or if I wanna maintain the equality, I could just subtract 16įrom the left-hand side. To the right-hand side so both sides have 16 added to it. On the left-hand side, I could either add that So let me do that in that blue color so we can keep track. And then once again, half of negative eight is negative four. If I look at this part, right over there, I could say x squared minus eight x. Actually, let me just do that really fast. If we complete the square, we're going to see something All I did is I subtractedĨ5 from both sides of this equation to get Would get x squared minus eight x minus 84 is equal to zero. Some people feel moreĬomfortable solving quadratics if they have that quadraticĮxpression be equal to zero. We could, right from the get-go, try to subtract 85 from both sides. Or another way of thinking about it, I could write it as x is equal to, four plus 10 is 14. And then what do I get? I get x is equal to four plus or minus 10. And now I just add four toīoth sides of the equation. So x minus four couldīe either one of those. All I did is took the plus or minus square root of a hundred. X minus four, is equal to positive or negative 10. Now if something squared is equal to 100, that means that the something is equal to the positive or the negative Four squared, these cancel out, is going to be equal to 100. And we are left with x minus four squared. And the easiest way weĬan do that is subtract one from both sides. And now we wanna get rid of this one on the left-hand side. And then we have plus one is going to be equal to, what's 85 plus 16? That is 101. And you can verify, x minusįour times x minus four is, indeed, equal to this. This is the same thingĪs x minus four squared. And why was that useful? Well, now what I've just put in parentheses is a perfect square. Notice, I've just done the same thing to both sides of this equation. And if I want, I could then subtract a 16 from the left-hand side. So I'm gonna add positiveġ6 on the left-hand side. Of negative eight." That would be negative four. Negative eight coefficient on the first-degree term, I could say, "Okay, let me take half Of the left-hand expression "a perfect square?" Well, if I look at this Sides of this equation "that could make this part Now if I wanna complete the square, I just have to think, "What can I add to both And then I have, I'll write the plus one out here, is equal to 85. So to do that, let me write it this way, x squared minus eight x. So one technique could be just, let's just try to complete the square here on the left-hand side. What x values satisfy the equation? All right, now let's work And what I want you toĭo is pause this video and see if you can solve it. We restate the patterns here for reference.Given this equation here. Let’s look at two examples to help us recognize the patterns. ![]() What happens if the variable is not part of a perfect square? Can we use algebra to make a perfect square? We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form ( x − k ) 2 ( x − k ) 2 in order to use the Square Root Property. In the last section, we were able to use the Square Root Property to solve the equation ( y − 7) 2 = 12 because the left side was a perfect square. \)Ĭomplete the Square of a Binomial Expression ![]()
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