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Solve the equation 𝑥 minus one multiplied by 𝑥 plus six multiplied by 𝑥 minus four multiplied by 𝑥 plus seven equals zero.
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So you might think, well, what do we need to do?
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Do we need to start distributing across our parentheses.
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But in fact, no, cause what we have is an equation in fully factored form.
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So therefore, finding the solutions is fairly simple.
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Well, all we need to do is take a look at our parentheses and see which 𝑥-values are gonna make them equal to zero.
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And that’s because if one of them is equal to zero, that means the answer to the left-hand side is gonna be equal to zero because zero multiplied by anything is zero.
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And on the right-hand side of the equation, we have zero, and that wants to be our solution.
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So what we can do is equate them to zero, so if we’ve got 𝑥 minus one equals zero, well, then the value of 𝑥 that’s gonna make that parentheses equal to zero is gonna be one.
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So that’s our first solution.
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Then for our second solution, what we’ve got is 𝑥 plus six equals zero.
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So then if we subtract six from each side of the equation, we’re gonna get 𝑥 equals negative six.
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So that’s our second solution.
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And then, for our third solution, we take a look at the parentheses where we’ve got 𝑥 minus four.
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And then if we equate that to equal zero, then we just add four to each side, so we get 𝑥 equals four.
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So then we use the same method for our final solution, so we get 𝑥 equals negative seven.
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So therefore, we can say the solutions are 𝑥 equals one, 𝑥 equals negative six, 𝑥 equals four, and 𝑥 equals negative seven.
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And for a quick tip to solve a question like this nice and easily, you might’ve noticed that, in fact, the values of 𝑥 were just the opposite sign to the value that we had inside the parentheses.
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So we had plus one or positive one instead of negative one, negative six instead of positive six, positive four instead of negative four, and negative seven instead of positive seven.
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And although this is a very nice and quick, easy way to solve a problem like this, it is worth mentioning it is slightly different if we have something like this.
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So here, we’ve got two 𝑥 plus three.
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So if this was our parentheses, then we have to equate this to zero.
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Well, if we’ve got two 𝑥 plus three equals zero and we subtract three from each side, we get two 𝑥 equals negative three.
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And then to work out what 𝑥 is, we divide by two.
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So, in fact, what we have is 𝑥 is equal to negative three over two.
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So it’s not just negative three; it’s negative three over two.
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And that’s because the coefficient of 𝑥 was greater than one.
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So be careful if you have something like this.