TargetIIT

[im]http://codecogs.izyba.com/gif.latex?lets%20%5C%20consider%20%5C%20%5C%5C%20P%28x%29%3D4x%5E2%28x%5E4%20-%5Cfrac%7B3%7D%7B2%7D%29%20%5C%5C%20P%28x%29%3D4x%5E2%28x+%28%5Cfrac%7B3%7D%7B2%7D%29%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%29%28x-%28%5Cfrac%7B3%7D%7B2%7D%29%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%29%28x%5E2+%28%5Cfrac%7B3%7D%7B2%7D%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%29%20%5C%5C%20roots%20%5C%20are%20%5C%200%2C0%20%2C+%28%5Cfrac%7B3%7D%7B2%7D%29%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D%2C-%28%5Cfrac%7B3%7D%7B2%7D%29%5E%7B%5Cfrac%7B1%7D%7B4%7D%7D[/im]
[image]39424857.jpg[/image]
[im]http://codecogs.izyba.com/gif.latex?now%20\%20putting%20\%20P%27%28x%29%20%3D0%20\%20we%20\%20can%20\%20find%20\%20those%20\%20peaks%20\\%20P%27%28x%29%3D24x^5%20-12%20x%3D0%20\\%20x%3D0%20\%20is%20\%20obvious%20\%20from%20\%20graph%20\\%20x_0%20%3D%28\frac{1}{2}%29^{\frac{1}{4}}%20\\%20P%28x_0%29%3D%20\sqrt{2}%20-%203\sqrt{2}%20%3D-2\sqrt{2}%20\\%20now%20\%20adding%20\%20a%20constant\%20means%20\%20dragging%20\%20the%20\%20graph%20\%20along%20\%20y%20\%20axis%20\\%20so%20\%20we%20\%20see%20\%20exactly%20\%20two%20\%20roots%20\%20are%20\%20there[/im]
[image]39424353.jpg[/image]
[im]http://codecogs.izyba.com/gif.latex?roots%20\%20are%20\\%20(\frac{1}{2})^{\frac{1}{4}},-(\frac{1}{2})^{\frac{1}{4}}[/im]

I thought of the substitution

[im]http://codecogs.izyba.com/gif.latex?x%5E2%3Da[/im]...

So on dividng by 2, we finally have [im]http://codecogs.izyba.com/gif.latex?2a%5E3-3a+%5Csqrt%7B2%7D%3D0[/im]

From which the factorisation [im]http://codecogs.izyba.com/gif.latex?%282a%5E2-2%5Csqrt%7B2%7Da+1%29%28a+%5Csqrt%7B2%7D%29%3D0[/im] follows.

[im]http://codecogs.izyba.com/gif.latex?%28%5Csqrt%7B2%7D-a%29%5E2%3D0%20%5CRightarrow%20x%3D%5Csqrt%5B4%5D%7B2%7D[/im]

Also complex roots are [im]http://codecogs.izyba.com/gif.latex?%5Csqrt%5B4%5D%7B2%7Di[/im]...

Ok, I had not thought of that.

But moving on from Soumik's first substitution x[p]2[/p]=a,

we get [im]http://codecogs.izyba.com/gif.latex?2a%5E3-3a+%5Csqrt%202%20%3D%200[/im]

Now, when you see a[p]3[/p] and 3a etc. , there's just the possibility that a sin 3x substitution is possible.

So letting [im]http://codecogs.izyba.com/gif.latex?a%20%3D%20%5Csqrt%202%20%5Csin%20y[/im] and dividing by √2 throughout you get

sin 3y = 1

and the roots are then got very easily