Solving a cubic equation and looking for integer solutions. Substitution rules.

This video is about a cubic equation that can be solved without using the cubic formula.

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[hamidkh]

a very nice method. Thank you professor

[scottleung]

I just found x directly using simple arithmetic - I didn't see the point in substituting.

[moeberry]

I let u= the average of x+2, x+1 and x+3, which gives x+2, which makes the equation a little simpler.

[kaslircribs]

Beautiful way of solving an equation! Thank you again sir!

[nalapurraghavendrarao]

Students who have solved lots of problems in mensuration specifically volumes of spheres are aware about these integers., that sum of cubes of 3, 4, 5 is=cube of 6.

[arpansit]

Sir i think it can be done in another way since (x+3) ^3 can be written as (x+2+1) ^3 so by expanding we get (x+2) ^3 +1 +3(x+2) (x+3) then in the lhs (x+2) ^3 gets ruled out then we are left with x^3 + (x+1) ^3 = 3(x+2) (x+3) +1 then by expanding some terms get cancel out and we end up with a cubic solution where 3 can be a solutions and -3+-i√3/2 are the complex solutions.

[michaelempeigne]

i found a nice way to do the problem for 2, 5, 11, 23, 47, using finite differences and some common formulas :


taking the differences yields : 3, 6, 12, 24, and this sequences has a formula of 3*2^(n -1 )

but each term is one less than all the terms in this sequence so the formula would be 3*2^( n -1 ) - 1

Now if we rewrite it : ( 2 + 1 ) * 2^(n - 1 ) - 1 = 2^n + 2^( n -1 ) - 1

[shanmugasundaram]

The substitution x=u+3 does the magic.