Solving a 'Stanford' University entrance exam

[00:00:00]Welcome to Smart Math. Today, I'm trying to solve this math Olympic problem with a very easy way. So, our equation is x cubed minus two divided by by three, that is equals to square cube root 3x + 2.

[00:00:15]Find the value of x where x is or a real number.

[00:00:20]So, if I'm let y is equals to y is equals to cube root of 3x + 2.

[00:00:31]Cube root here. So, that So, if I'm again further solve if I'm taking both side by cube.

[00:00:40]If I'm taking both side by cube here and cube here, so this and this eliminate.

[00:00:45]So, our equation become y cube, that is equals to 3x + 2x + 2.

[00:00:53]So, this two is positive goes equal opposite side means sign change.

[00:00:57]So, y cube minus two that become three of x and that is our equation number one.

[00:01:06]So, now >> [snorts] >> So, now x cubed minus two divided by by three that is equals to y here.

[00:01:20]So, [snorts] this mean this three goes opposite side mean 3y.

[00:01:28]So, x cubed minus two that become 3y here. So, multiply this down.

[00:01:35]So, if I'm this is called equation number two.

[00:01:42]So, [snorts] if I'm out equation number one here, so our equation become equation number one is y cube minus two that is equals to three of x.

[00:01:55]So, now if I'm separate out by minus term here, so our equation is become this way 2 and plus 2 is cancel. So this term in plus minus is here plus minus here. So our remainder term is x cubed minus y cubed that is equals to 3y minus 3 of x here.

[00:02:19]So our next step is become if [snorts] I'm take take common here minus here. So our equation become this is a negative become positive 3 of x and positive become negative 3 of y here.

[00:02:33]x cubed minus y cubed equals So this is this is >> [snorts] >> if I'm taking 3 here common minus 3 common here. So our equation [snorts] become x minus y that is x cubed minus y cubed.

[00:02:50]So further next step become this is negative term goes equal opposite side mean positive term. So equation become x cubed minus [snorts] y cubed plus 3 into x minus y that is equals to 0. So this is a squared a a cubed minus b cubed formula they will be equal to a minus b into a squared plus ab plus b squared.

[00:03:29]>> [snorts] >> So this equation become >> [snorts] >> x minus y into x squared plus xy plus y squared here.

[00:03:44]>> [snorts] >> plus 3 into x minus y that is equals to 0.

[00:03:52]So if I'm taking minus X minus Y common here, common here and common here. So, the equation become X minus Y into X squared plus XY plus Y squared and remainder term here is three.

[00:04:13]That That is equals to zero.

[00:04:16]So, we have two two factors and X minus Y This is equals to zero.

[00:04:26]Or X squared plus XY plus Y squared plus three that is equals to zero.

[00:04:35]So, So, now we have two cases. This is case number one and this is case number two. So, case number one, this is negative term goes opposite side mean positive.

[00:04:50]X is equals to Y.

[00:04:53]X is equals to Y here. So, if I'm trying to solve this case, so our equation is X X squared plus XY plus Y squared plus three here.

[00:05:03]So, we have uh taking uh X squared plus XY plus Y squared here.

[00:05:13]And uh if we add Y divided by by two to completing square method minus Y divided by by two here.

[00:05:24]That is equals to zero.

[00:05:26]>> [snorts] >> So, if if I'm rearrange that the equation equation become X squared plus two into X into two into X into Y divided by by two plus Y divided by by 2 whole square plus y square here minus y square divided by by 4 plus 3 that is equals to 0.

[00:06:00]So, this is this is xy this term is xy and we both multiply 2 2 cancel out, they will be same xy.

[00:06:10]So, this equation look like this equation look like the formula of a square plus 2ab plus b square formula that is becomes a plus b whole square.

[00:06:26]So, our equation become >> [snorts] >> x plus y divided by by 2 square plus 4y here.

[00:06:43]We take a now take take a LCM here. They now take the these two term take LCM.

[00:06:51]So, 4y 4 1 into 4 is 4 4 into y square 4y square minus y square here plus 3 that is equals to 0.

[00:07:01]So, this term look like x plus y divided by by 2 whole square plus 4 minus 1 is 3y square divided by by 4 plus 3 that is equals to 0.

[00:07:19]So, [snorts] this look like the formula value of x is the value of x is greater than or greater or [snorts] equal to 0. This is look like also greater or equal to 0. And this is not not not a constant it's just a constant number that is equal to not a the equal to 0. This is greater than zero. So, we got the imaginary value.

[00:07:46]So, this is they have no real solution.

[00:07:53]So, now I'm considering the case number one that is x is equals to y here.

[00:08:00]x is equals to y here. So, now I'm recall the value of x squared minus two x squared minus two divided by by three that is equals to y here. So, x is equals to y. So, now if I'm this three goes that side mean multiply here. So, equation become x cubed here, not x squared. x squared x cubed minus two that is equals to three y here. So, this is three y here. So, now as our case number one is x is equals to y. So, now replace x is equals to y here. So, our equation become x cubed minus two that is equals to three of x here. So, this is positive goes equal opposite side mean negative.

[00:08:55]So, equation become x cubed minus three of x minus two that is equals to zero.

[00:09:04]So, now if I'm consider the value of x that is plus minus one plus minus two plus minus three. So, up to so on. So, now if I'm trying to solve the equation become at x is equals to minus one. Is this value possible? Is this the value of is a solution or not? To put in this main equation. So, equation become minus one here.

[00:09:30]x replace x with minus one here. minus one x cubed minus three into minus one minus two that is equals to zero.

[00:09:39]So, that is equals to zero. Minus minus plus, plus one, and again minus one here is equals minus one. Minus minus plus three ones are three. Minus two, that is equals to zero. So, this is three. Minus minus plus, minus three here.

[00:09:57]Minus three plus one here is equals to zero. So, zero is equals to zero. This prove and x that is equals to minus one is a solution possible.

[00:10:13]So, x plus one is a factor. So, x plus one is a factor.

[00:10:24]So, now I put now if I put in main equation, so equation become x cube plus x square here.

[00:10:37]>> [snorts] >> So, if I'm trying to x square here because equation is not a not a This is the main equation. X cube minus three of x minus three [snorts] of x plus minus two that is equals to zero.

[00:10:53]So, now if I'm if I'm trying to solve this equation, x cube So, they they got no nothing have. If I'm all plus square and minus x square here, the equation will be same. So, this is minus three here. So, this is minus three here. If I'm loading here minus x into minus two is become minus three x.

[00:11:17]So, our remainder term is minus two, that is equals to zero. So, now if I'm common here x square, so equation become x square x plus one.

[00:11:29]So, if I'm common here minus x here, minus x here x plus one equals equation. If I'm common minus two here. Minus two here, equation become X plus one that is equals to zero.

[00:11:43]So, if I'm common again here, equation X plus one, X plus one, X plus one. The equation become X plus one into remainder is X squared minus X minus two.

[00:11:59]That is equals to zero.

[00:12:01]So, X plus one is equals to zero or X squared minus X minus two that is equals to zero.

[00:12:11]So, this is positive goes equal opposite side mean negative.

[00:12:14]>> [snorts] >> So, X is equals to minus one here.

[00:12:18]So, X is equals to minus one here.

[00:12:20]So, now this is the pure quadratic equation. To solve this quadratic equation with with the to solve this quadratic equation with factorization method.

[00:12:32]So, equation become X squared is a factor is minus if I'm uh put minus two plus X become minus X and one one into two is minus two here.

[00:12:45]Minus two that is equals to zero. So, X squared if I'm common if I'm common No, wait a second. If I'm common here X So, equation is X minus two. If I'm common plus one here.

[00:13:02]Equation is X minus two that is equals to zero.

[00:13:05]So, if I'm common again X minus two, X minus two here.

[00:13:09]X minus two is is a factor and X remainder term is X plus one is a factor. So, uh if I if I'm solve this equation X minus two that is equals to zero or X plus one that is equals to zero. This minus goes opposite side and positive and X is value of X is two.

[00:13:31]And this X value goes opposite sign, mean minus one here.

[00:13:37]So, [snorts] the value of X is two and minus one is our final answer.

[00:13:43]So, thanks for watching. Keep like, share, and subscribe.