Deductive reasoning

so they tell us that here I'm solved the equation five plus the square root of x plus 14 is equal to X plus seven using the following step so let's see what he did he subtracted 5 from both sides yea 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 used he uses the pattern for square binomials to expand the right hand side okay so he just multiplied out X plus 2 times X plus 2 to get x squared plus 4 X plus 4 then he subtracted X plus 14 from both sides so you guys gets a zero on the left hand side so when you take X from for X you get 3x when you take 14 from 4 you get negative 10 so that all makes sense then you factor the right-hand side let's see 5 times negative 2 is negative 10 5 plus negative 2 is 3 that makes sense then it uses a zero product property to solve the equation that's just this property that look if you have two things and you take their product and it equals zero one or both of them must be equal to zero so that means X plus 5 is equal to zero or X minus 2 is equal to zero and so if X plus 5 is equal to zero that's X is equal to negative 5 X is equal to 2 is zero 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 zero or X minus 2 is equal to zero 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 you substitute 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 the square root of 9 this becomes 2 and 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 to 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 were talking about so we're taking the positive square root and 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 the 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's starting with something he knows it's true and gets to something else he knows it's true and this is a bit of review you're probably wondering why this why the negative 5 wasn't a solution and it's 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 that's what 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 have 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've explained it before in the past so this was deductive reasoning