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Product of two numbers to equal zero without at least one of them being equal to zero? And the simple answer is no. If I had two variables, let's say A and B, and I told you A times B is equal to zero. This is expression is being multiplied by X plus four, and to get it to be equal to zero, one or both of these expressions needs to be equal to zero. Thing being multiplied is two X minus one. Things being multiplied, and it's being equal to zero. So at first, you might be tempted to multiply these things out, or there's multiple ways that you might have tried to approach it, but the key realization here is that you have two Satisfy this equation, essentially our solutions If you can figure out the X values that would And we're done.That we've got the equation two X minus one times X plus four is equal to zero. y is equal to negativeĢ/3 right over there. And so you get y is equal to negative 2/3. To 1/3 using this equation? We could have used the original one, but this is even simpler. So we could figure out what is x when, or what is y when x is equal And I like to focus on the simpler of the two equations. To be the corresponding y we get for that x in either equation. But we still have toįigure out the y value. But now we figured out the x value of the other solution. So we figured out the, we already saw the solution You get three x is equal to one, or x is equal to 1/3. And for three x minus one equals zero, add one to both sides. Minus two equals zero is when x is equal to two. So x minus two could be equal to zero, or three x minus one is equal to zero. And so a solution would be a situation where either of these is equal to zero, or, I'll scroll down a little bit more. So I have zero is equal to, if I factor out an x minus two, I'm going to get x minus I'll scroll down a littleīit so I have some space. And then I can factor out a negative two. In these second two I canįactor out a negative one, so I have negative one times x minus two. X times x minus two, and in these second two I can factor out. So then zero is equal to, so if I group these first two I can factor out a three x. You could also use the quadratic formula. I'm just factoring by grouping, for those of you are notįamiliar with this technique. Rewrite this whole thing as zero is equal to three x squared, and then instead of negative seven x, I can write negative six x and minus x. And can I think of those same two, a plus b, where it's going to be equal to negative seven? And actually, negative This looks unfamiliar, you can review factoring by grouping.
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Product of three and two? Three times two, and if Obvious way to factor it? Can I think of two numbers, a times b, that's equal to the We get three x squared minus seven x plus two. And then whatta we get? On the left-hand side we just get zero. Now we wanna get a zero one side of this equation, so let's subtract x. X minus one is equal to three x squared minus six x plus one. Substitute x minus one back in for y, and so we get The stuff cancels out, is equal to x minus one. And so we are going to get y, and then all the rest of Y is to add x to both sides and subtract one from both sides. And actually, let me color code it, because this one is in red,Īnd this one is the line in that blue color. Into my quadratic one, and then hopefully I can solve for x. This linear equation as in terms of y, if I can solve for y, then I can substitute what y equals back into my first equation,
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Well one way to tackle,Īnd this is one way to tackle any system of equations, And our next equation right over here, y minus x plus one is equal to zero. The equations is not linear, where it is a quadratic. Well to do that, we're going to have to try to solve this system of equations, and this is interestingīecause this is a system of equations where one of Out this intersection point right over here. That sits on the graph of both of these curves, that means that it satisfies both of these equations, that it's a solution toīoth of these equations. These intersection points is, because it's a point This second one seems clearly identifiable because, when I look at the grid, it looks clearly to be at a value of x equals two and y equals one. I see that one, and I see that one there. Point is clearly identifiable from the graph. If you were doing it on Khan Academy, you would type it in. Screenshot from the exercise on Khan Academy, but I'm just going to write on it. What is it? And they want us to put it in here. And the first thing they ask us is one intersection point So we can see the parabola here in red, and we can see the line here in blue. Squared minus six x plus one and the line given by y minus x plus one equals zero are graphed. Told the parabola given by y is equal to three x