Topic A: Lessons 6-7: Solving basic one-variable quadratic equations
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Use the cards below to create a list of steps in order that will solve the following equation. 3 times x plus 6 squared is equal to 75. And I encourage you to pause this video now and try to figure it out on your own. Figure out which of these steps and in what order you would do to solve for x here. So I'm assuming you've given it a go. So let's try to work through it together. And first, let me just rewrite the equation. So we have 3 times the quantity x plus 6 squared is equal to 75. So what I want to do is I want to isolate the x plus 6 squared on the left-hand side. Or another way of thinking about it-- I don't want this 3 here anymore. So how would I get rid of that 3? Well, I could divide the left-hand side by 3. But if I do that to only one side of the equation, it won't be equal anymore. These two things in yellow were equal to each other. If I want the equalities to hold, anything that I do to the left-hand side, I have to do the right-hand side. So let me divide that by 3 as well. And so on the left-hand side, I am left with x plus 6 squared is equal to 75 divided by 3. So 75 divided by 3 is 25. So actually, let me just pick out the first one I did. I divided both sides by 3. So that was my first step then. Let me write that in a darker color. So that was my first step right over there. Now let's think about what we're doing. We're saying that something squared is equal to 25. So this something could be the positive or negative square root of 25. So we could write this as x plus 6 is equal to the plus or minus square root of 25. So I'm essentially taking the positive and negative square root of both sides. So, let's see. This looks like this step. I took the square root of both sides. That's step number two. And so, let me just rewrite this. This is the same thing as x plus 6 is equal to plus or minus 5. And now I want to just have an x on the left-hand side. I want to solve for x. That's the goal from the beginning. So I would like to get rid of this 6. Well, the easiest way to do that is to subtract 6 from the left-hand side. But just like before, I can't just do it from one side of an equation. Then the equality wouldn't be true. We're literally saying that x plus 6 is equal to plus or minus 5. So x plus 6 minus 6 is going to be equal to plus or minus 5 minus 6. Or actually, let me write it this way. So let me subtract 6 from both sides. On the left-hand side, I'm left with an x. And on the right-hand side, I could write it this way. Let me do it in that green color. I have negative 6 plus or minus 5. So what are the possible values of x? Or actually, I keep forgetting. We don't have to actually give the value for x. We just have to say what steps we did. So then, let's see. After we took the square root of both sides, we then subtracted 6 from both sides. So that was step three right over there. Then that got us to essentially the two possible x's that would satisfy this equation right over here. And just for fun, let's actually solve it all the way. So if we solve it all the way, so x is equal to negative 6 plus 5 is negative 1, or x is equal to negative 6 minus 5 is negative 11. And you could verify that both of these work. If you put either of them in here-- if you put negative 1 here, you get negative 1 plus 6 squared is 5 squared. If you put negative 11 here, it's negative 11 plus 6 is negative 5 squared. Obviously either plus or minus 5 squared is going to be 25. 25 times 3 is 75. So these are our three steps. We divided both sides by 3. Then we took the square root of both sides. Then we subtracted 6 from both sides. And then we were essentially done. So let's input those steps. So the first thing we did, we divide both sides by 3. That's the first thing we did. And then we took the square root of both sides. And then we subtracted 6 from both sides. We got it right.