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## Algebra (all content)

# Deductive reasoning

Sal analyzes a solution of a mathematical problem to determine whether it uses deductive reasoning. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- Will there be reasoning which both has deductive and inductive? How to call this?(7 votes)
- I'm not exactly sure, but I'm guessing that as soon as you put something into your reasoning that isn't a fact, it can't be called deductive anymore.(5 votes)

- I didn't understand what Sal said at3:33that by taking the negative square root we would have satisfied the equation with -5.

But how is that possible since the Square Root of a Negative number does not exist, it's an imaginary number!(2 votes)- He is talking about the negative square root not the square root of a negative; there is a difference. He is talking about taking the square root of the positive number, then flipping the sign so the answer is negative. You are right to be confused though, because -(sqrt)() looks a lot like (sqrt(-).(7 votes)

- What grade level is this meant to be for?

Because I'm in Grade 6 and I can do Algebra, but I find this stuff confusing.

Not the squaring and adding , but the whole concept and like, the 'Zero Product Property'.

What is the that and should I be learning this right now?(2 votes)- This is in the pre-calculus playlist, and you do calculus in college. So this can be considered high school level. The zero product property simply says that when two numbers multiplied give the value 0 then one of them must be 0. Because only 0*x or x*0 or 0*0 can be 0. So if (x+5)*(x-2) = 0 then, at least one of them must be 0. And we take each case, for each possible solution. x+5 = 0 so x = -5 or x-2 = 0 so x = 2. This is useful if you are doing second degree equations (a*x^2 +b* x + c = 0, which have 2 solutions).(5 votes)

- At2:05, why does he only use the principle root?(3 votes)
- Generally speaking, we always expect a square root of a number to be positive.(2 votes)

- When you solve for example:
`x^2=4`

you get`x=+-2`

+- is so you don't lose an answer. When you have equation like`sqrt(x+14) = x+2`

you have to set limits before squaring them so you don't gain an answer. Thus`x>=-14`

and`x>=-2`

.(3 votes) - On the third line, I don't understand how he got x^2+4x+4.(2 votes)
- Hi ahewatt,

On the second line both sides of the equation were squared and results in (x+2)^2 on the right side of the equation.

(x+2)^2 = (x+2)(x+2)

Sal explains how to expand similar binomial multiplication in another video (https://www.khanacademy.org/math/algebra-basics/quadratics-polynomials-topic/multiplying-binomials-core-algebra/v/multiplication-of-polynomials) and the first binomial multiplication problem begins approximately at3:10.

Hope this was helpful!(3 votes)

- how do you end up with x=2 when you got 9 do you have to do the false version(2 votes)
- What is deductive resioning(1 vote)
- Deductive reasoning is using facts to come up with more facts. No generalizations or assumptions are made. =D(1 vote)

- I watch to first 2 videos of deductive and inductive resoning. Should this topic be in statistics(1 vote)
- Is there a way to practice using deductive reasoning in order to get better at it? We have to use this next year in geometry and I'm not very good at it.(1 vote)

## 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.