document.write( "Question 1164447: The problem is 8x^3=125. Why can't I simply take the cubed root of both sides to get 2x=5, and from there get x=5/2?\r
\n" ); document.write( "\n" ); document.write( "The solutions I have seen require I move all terms to the left and set the equation equal to zero, factor using difference of cubes formula, and solve from there, giving me x= 5/2, and x=-5+/-5i(square root of 3) all over 4. I don't understand why I can't just solve it as the equation is given, instead of setting everything to 0. Thank you! \n" ); document.write( "
Algebra.Com's Answer #789085 by greenestamps(13203)\"\" \"About 
You can put this solution on YOUR website!


\n" ); document.write( "Let's look at a simpler, more familiar example, to see what the problem is with just taking the cube root on both sides.

\n" ); document.write( "Suppose the equation is

\n" ); document.write( "\"x%5E2+=+4\"

\n" ); document.write( "If you just take the square root of both sides, you get x=2. But you know there are two solutions, 2 and -2.

\n" ); document.write( "Algebraically, to get all the solutions (in this case, both), you need to set everything equal to 0 and solve.

\n" ); document.write( "\"x%5E2-4=0\"
\n" ); document.write( "\"x-2%29%28x%2B2%29+=+0\"

\n" ); document.write( "x=2 OR x=-2.

\n" ); document.write( "In that case, both solutions are real numbers. In cases of equations with higher powers of x, that is often/usually not the case. A polynomial equation of degree n has n roots; only in rare special cases are all the roots real.

\n" ); document.write( "In your example, of the form x^3=A where A is any real number, there is always only one real root.

\n" ); document.write( "So when you solve the equation by taking the cube root of both sides, you find only the real root; and you have lost the information you need to find the non-real roots.

\n" ); document.write( "To find all three roots, you need to use the pattern for factoring the difference of cubes. That will give you a linear factor which gives a real solution and a quadratic factor that gives two non-real solutions.

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