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Zarrin · May 23, 2007, 03:11 PM

This is the problem:
Roots x of the equation 512x^3 - 4800x^2 + kx - 5625 = 0 are in arithmetic progression. Find the Largest of the three roots and express your answer as an exact decimal.

I've been trying this for hours but cannot figure it out, if someone could help it would be greatly appreciated!!

Thank you

galactus · May 23, 2007, 07:12 PM

Since the roots are in an arithmetic progression, they have a common difference.

Since the coefficient of x^3 is 512, then we must have:

$(8x+a)(8x+b)(8x+c)=0$

Expand out and this gives:

$512x^{3}+64x^{2}b+64x^{2}a+64x^{2}c+8xab+8xbc+8xac +abc=0$

By grouping, factoring and equating coefficients, we get:

$64(a+b+c)=-4800------>a+b+c=-75$

$8(ab+bc+ac)=k$

$abc=-5625$

What three numbers when multiplied equal -5625 and when added equal -75?

Hmmmm... how about -5, -25, -45. They have a common difference of 20.

Therefore we have:

$(8x-5)(8x-25)(8x-45)=0$

You can use those in the equation for k to find k.