MHB What are the Values of a, b, and c in the Real Root Equation 8x^3-3x^2-3x-1=0?

Thread starter anemone
Start date Jun 19, 2013

AI Thread Summary

The equation 8x^3 - 3x^2 - 3x - 1 = 0 has a real root expressed as (√[3]{a} + √[3]{b} + 1) / c, with a, b, and c being positive integers. Through manipulation, it is found that a = 81, b = 9, and c = 8. The sum of these values, a + b + c, equals 98. The solution process involves rewriting the equation and factorization. Thus, the final answer is 98.

Jun 19, 2013

anemone

Gold Member

MHB

POTW Director

The real root of the equation $$8x^3-3x^2-3x-1=0$$ can be written in the form $$\frac{\sqrt[3]{a}+\sqrt[3]{b}+1}{c}$$, where $$a,\;b,\,c$$ are positive integers.

Find $$a+b+c$$.

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Jun 19, 2013

mathworker

$$8x^3-3x^2-3x-1=0$$
$$9x^3=(x+1)^3$$
$$9=(1+\frac{1}{x})^3$$
$$x=\frac{1}{\sqrt[3]{9}-1}$$
by factorizing,
$$a=81,b=9,c=8$$
$$a+b+c=98$$

Last edited: Jun 19, 2013

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