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Find how many roots of the equations x^4...

Find how many roots of the equations `x^4+2x^2-8x+3=0.`

Text Solution

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Let `f(x) =x^(4)+2x^(2)-8x+3`
`f(x)=4x^(3)+4x-8=4(x-1)(x^(2)+x+2)`
Now f(x)=0`rArr`x=1
Hence graph of y = f(x) has only one turn (maxima/minima)
Now `f(1)=1+2-8+3lt0`
Also `underset(xrarr00)lim f(x)=00`
The graph of the function is as shown in the following figure From the graph equation has two real roots.
Also ,`alpha` in (0,1) ..`[alpha]=0`
`f(2)lt0`
`betain (1,2)..[beta]=1`
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