We can open these brackets X to the power of three plus 4 X #### square

so the volume of this box is and you can say that X #### square

is the area of the base multiplied by the height of X to the power of three plus 4 X #### square

in either direction if that Volume of the box 225 centimeters is cube, what are its ### dimensions

, so the volume is 225 less, replacing V by X is 225, and then we have that it should be equal to X to the power of three plus 4 x #### square

d. We need to ## find

the value of x so we take away 225 from both sides we get 0 equals 2x cube plus 4x #### square

minus 225 Now to ## find

the zeros for this ### polynomial

we should drive what are factors of 225.

A good value to try is five so let's call this a ### polynomial

so for this ### polynomial

try the value five so let's try you can take a calculator and then calculate that 5q plus four times five #### square

d minus 225 equals which gives us five cubic plus four times five #### square

d minus 225 which is zero so we get a zero which means X minus five is a correct factor to ## find

the rest of the factors we can actually do a synthetic division and ## find

the #### equation

such that we have at least one value of x and that gives us x equals five can give a range of solutions if i replace x with 5 then i get longitude and latitude is 5 and the height is 5 plus 4 which is 9 units cents well the question is almost complete at this point we can also write down our answer and our answer is that the ### dimensions

5cm x 5cm x 9cm s ind, so the correct size of the box, but there's no harm in continuing here, since we know that one of the factors is 5, we can do synthetic division or long division, and here we have our #### equation

, which are the coefficients 1 that's 4x cubic x #### square

it's 4 let's substitute 0 for X and we have minus 225 3 5 is a factor I should say X minus 5 is a factor so we can divide by 5 what do we get 1 take down multiply by five so we get five five plus four is nine five times nine is forty five add forty five times five to twenty five so we actually get zero as expected which means the ### polynomial

hora can be written as so we can use the ### polynomial

function can be written as equal.

I mean this point of my question correctly as X minus five times that is #### square

plus 9x plus forty five by #### equation

now we c ## find

out if there are more solutions to this okay which means more factors of this quadratic #### equation

. Now when you try to ## solve

this quadratic #### equation

we use the quadratic #### equation

as x equals two minus P which is minus 9 plus or minus B #### square

which is 9 #### square

d is 81 minus 4 times AC 4 times 45 is 4 times 5 is 20 4 times 4 is 16 and 2 is 180 so here we get a negative term divided by 2 which means there are no more solutions that aren't actually correct so the only combination is the one we get and now we can write ours with certainty that the ### dimensions

are two inches by two inches by three inches.

Well, some of you at the stage may think that this was kind of a waste of time trying the other routes soon you will come across questions in the test where the wording is different. They say what possible ### dimensions

are, what all possible ### dimensions

are. In some of these cases you might ## find

some real roots and different combinations of solutions, so the idea to take this is even if you got an answer. Try to see the others. You have a quality #### equation

that's so easy to keep solving and there's a chance you'll get more than one solution, so move on to the next question.

I hope you appreciate it. Thank you very much and good luck

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