What mistake did Mindy make when solving this quadratic equation?

[halvizo1031]

Homework Statement

Mindy solves the problem (x+4)(x-3)=8 by setting x+4=8 and x-3=8, thereby getting the solutin x=4 from both equations. she checks her answer by substituting x=4 into the original equation and finds that it works. She concludes that this quadratic equation has only one solution. if we check x=-5, it also works. she lost a solution. WHAT MISTAKE DID MINDY MAKE? WHAT MIGHT SHE NOT UNDERSTAND? WHAT PROPERTY OF FIELDS/INTEGRAL DOMAINS IS MINDY'S MISTAKE RELATED TO?

Homework Equations

The Attempt at a Solution

If she were to first FOIL, subtract 8 from both sides, and solve for x, then she would get both solutions. BUT, i am stuck in explaining what she might not understand and what fields/integral domains is her mistake related to...

 

[QuarkCharmer]

I'm thinking that this has to do with Zero Divisors. Specifically the nonexistence of them. Does that help?

 

[halvizo1031]

that makes sense as far as explaining her misunderstanding...but what about the property of fields/integral domains that her mistake is related to?

 

[HallsofIvy]

Mindy is assuming "if ab= c then a= c or b= c" which is not true. It is a mistaken version of the "zero product" rule that says "if ab= 0 then a= 0 or b= 0". That, in turn is true because if $a\ne 0$ we can divide by it getting b= 0 and the reverse. That is where "no zero divisors" in a field is applied and where Mindy's mistake is. "No zero divisors" does NOT mean "no c divisors" for c non-zero.

 

[ideasrule]

Homework Statement

Mindy solves the problem (x+4)(x-3)=8 by setting x+4=8 and x-3=8, thereby getting the solutin x=4 from both equations.

Um, you don't get x=4 from both equations. If x-3=8, x=11, not 4.

 

[HallsofIvy]

Um, you don't get x=4 from both equations. If x-3=8, x=11, not 4.

Yeah, but by this time Mindy is so lost, it doesn't matter!

 

[halvizo1031]

you are correct that she "should" get 4 and 11 but i guess it doesn't matter since her process and reasoning is incorrect. unless the professor made a mistake in typing this question...thank you both for your input.