Quadratic equations word problem: box dimensions

the volume of a box is 405 cube units or gets cubic units so they just want to keep it general it could have been in cubic feet or cubic meters or cubic centimeters or cubic miles who knows they just want to keep it as units keep it as general as possible the length is X units the width is X plus 4 units and the height is 9 units let me draw this box here so let me draw this box so let's let me draw a little box here so we have a nice little visualization so they tell us that the length is X maybe we could call it well it's called this the length right there they say the width is X plus 4 the width is X plus 4 and the height is 9 of this box in units what are the dimensions of the box well they also tell us that the volume is 405 so the volume 405 let me do it this way so if we wanted to calculate the volume over to be well it would be the width it would be X plus 4 times the length times X times 9 times 9 that's literally the volume of the box now they also tell us that the volume of the box is 405 cubic units is equal to 405 so now we just solve for X so what do we get here if we distribute this X into this X plus 4 actually if we distribute a 9 X let me just rewrite it this is the same thing as 9 x times X plus 4 is equal to 405 9 x times X is equal to 9 x squared 9 X times 4 is equal to 36 X is equal to 405 now we want our quadratic expression to be equal to 0 so let's subtract let's subtract 45 405 from both sides of this equation so when you do that your right hand side equals 0 and your left hand side is 9 x squared plus 36 X minus 405 now is there any common factor to the these numbers right here well let's see 405 four plus zero plus five is nine so that is divisible by nine so all of these are divisible by nine let's just let's just figure out what 405 divided by nine is so nine goes into 405 nine goes into forty four times four times nine is thirty-six subtract you get 45 nine goes into 45 five times five times nine is forty-five subtract you get zero so it goes forty five times so if we factor out a 9 here we get nine times x squared actually even better you don't even have to factor out a 9 to think about it you can divide both sides of this equation by nine so if you can divide all of the terms by nine it won't change the equation you're doing the same thing to both sides of the equation which we've learned long ago is a very valid thing to do so here you get x squared if you if you just had this expression here and someone were told you to factor it then you would have to factor out the nine but because this is an equation it equals zero let's just divide everything by nine it'll simplify things so you get x squared plus 4x minus 45 is equal to zero and now we can try to factor this right here and this is this fits the pattern where we don't have a leading one out here so we don't even have to do it by grouping you just have to think what two numbers when I take their product I get negative 45 and when I take their sum I get positive 4 therefore apart one has to be positive one has to be negative they're positive versions have to be for Part B when you take the sum you're really taking their difference because one of them is negative so let's think about it well if you have positive nine and negative five I think that'll work right positive nine plus negative five is four and when you take the product you get negative 45 so you have X plus nine times X minus five is equal to zero just factored it out and then we've seen this before if you have two numbers when you take their product that equals zero that means one of these numbers at least has to be equal to 0 so X so this means that X plus nine is equal to zero let me scroll down a little bit X plus nine is equal to zero or X minus five is equal to zero so if we subtract nine from this equation right there you get X is equal to negative nine or if you add five to both sides of this equation here you get X is equal to five so these are both possible values of X right here so the box if you take X is equal to negative nine well X equal to negative nine won't work because if you put negative nine here you're going to have a you're going to have a box that has a width of negative five a length of negative nine and a height of nine and if we're talking about our reality we don't have negative distances like this that can't be the length or the width so x equals negative nine isn't appropriate for this problem because in this problem we need to have positive dimensions so let's see for what happens with x equals five x equals five x plus four is nine and this dimension right here is going to be five and that seems pretty reasonable for our reality and let's verify that this does end up with the volume of 405 nine times five is forty five times nine is indeed 405 we just figure that out over here that 45 that 45 times nine is 405 so we're done