Deductive and inductive reasoning
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So they tell us that Hiram solved the equation, 5 plus the square root of x plus 14 is equal to x plus 7, using the following steps. Let's see what he did. He subtracted 5 from both sides. Yeah, that's reasonable enough. That got that 5 out of the way. And then 7 minus 5 is 2. And then he squared both sides. The square root of x plus 14 squared is just x plus 14. That makes sense. And x plus 2 squared is x plus 2 squared. And then he uses the pattern for square binomials to expand the right-hand side. OK, so he just multiplied out x plus 2, times x plus 2, to get x squared plus 4x, plus 4. Then he subtracted x plus 14 from both sides, so he gets a 0 on the left-hand side. So when you take x from 4x, you get 3x. When you take 14 from 4 you get negative 10. So that all makes sense. Then he factored the right-hand side. Let's see, 5 times negative 2 is negative 10. 5 plus negative 2 is 3. That makes sense. Then he uses a 0 product property to solve the equation. That's just this property-- look, if you have two things and you take their product and it equals 0, one or both of them must be equal to 0. So that means x plus 5 is equal to 0, or x minus 2 is equal to 0. And so if x plus 5 is equal to 0, that's x is equal to negative 5. x is equal to 2 of 0, that's x is equal to 2. Let me just write that down here. So all he did in this step is he says, x plus 5 is equal to 0. Or x minus 2 is equal to 0. From this you get this right over here, because you subtract 5 from both sides. And then from this you get that right over there. Then he checked both answers. He substituted negative 5 into the original equation. So he substituted negative 5 in there. It shows up twice. 5 plus the square root of negative 5, plus 14 is equal to negative 5 plus 7. Let's see what else he did. Then this becomes square root of 9. This becomes 2. Then you get 5 plus 3 is equal to 2, which is false. This is not true. And he wrote that down. And then he tried out 2. His other solution. When you substitute 2 you get 2 plus 14, which is 16. 2 plus 7 is 9. Square root of 16 is 4. And this is the principal root of 16 we're talking about, so we're taking the positive square root. And then 5 plus 4 is equal to 9. So this works out. And then he says that the answer is x equals 2, which is right. Now this whole exercise, all they want us to know is, is this an example of deductive reasoning? Explain. And it is an example of deductive reasoning. He started off with a known statement, with a known-- we could call that a known fact-- if we assume that that's a fact. He started off with that. And just doing logical operations, he was able to deduce, step by step, he was able to manipulate other truths. He started with a fact and using logical operations he was able to come up with other facts. And go all the way down here and then check his answers, and eventually come up with the notion that if this is true, then this must also be true. So that is deductive reasoning. You start with facts, use logical steps or operations, or logical reasoning to come up with other facts. He's not estimating. He's not generalizing. He's not assuming some trend will continue. He started with something he knows is true and gets to something else he knows is true. And this is a bit of a review. You're probably wondering why the negative 5 wasn't a solution. And as a little bit of a hint here-- and I'll let you think about why it didn't end up as a solution-- when we took the square root of 9 here, we took the positive square root. And when anyone just says a square root like that, that means the positive square root, the principal square root. But if we were to take the negative square root here, this equation would have held up. Because 5 plus negative 3 is equal to 2. So I'll let you think about at what step of this equation would a negative number have worked? And it has something to do with when we square both sides. But that has nothing to do with the actual question at hand, so I'll leave you there. We explained it before, in the past. So this was deductive reasoning.