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# Deductive reasoning

Video transcript

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.