document.write( "Question 352923: x^4 + radical 3 * x^2 - 3 = 0 \n" ); document.write( "

Algebra.Com's Answer #252315 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"x%5E4+%2B+sqrt%283%29+%2A+x%5E2+-+3+=+0\"$
\n" ); document.write( "The key to making this problem solvable is to recognize that the exponent on $\"x%5E4\"$ is twice as bug as the exponent on $\"x%5E2\"$ . This makes your equation in \"quadratic form\" for $\"x%5E2\"$ . So we can use techniques for solving quadratic equations to solve for $\"x%5E2\"$ !

\n" ); document.write( "To make this process clearer, I am going to use a temporary variable. Let q = $\"x%5E2\"$ . Then $\"q%5E2+=+%28x%5E2%29%5E2+=+x%5E4\"$ . Substituting these into the equation we get:
\n" ); document.write( " $\"q%5E2+%2B+sqrt%283%29%2Aq+-+3+=+0\"$
\n" ); document.write( "This is clearly a quadratic equation. With the square root in there, factoring will not be easy. So we'll use the Quadratic formula:
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\n" ); document.write( "Simplifying:
\n" ); document.write( " $\"q+=+%28-sqrt%283%29+%2B-+sqrt%283+-+4%281%29%28-3%29%29%29%2F2%281%29\"$
\n" ); document.write( " $\"q+=+%28-sqrt%283%29+%2B-+sqrt%283+%2B12%29%29%2F2\"$
\n" ); document.write( " $\"q+=+%28-sqrt%283%29+%2B-+sqrt%2815%29%29%2F2\"$
\n" ); document.write( "In long form:
\n" ); document.write( " $\"q+=+%28-sqrt%283%29+%2B+sqrt%2815%29%29%2F2\"$ or $\"q+=+%28-sqrt%283%29+-+sqrt%2815%29%29%2F2\"$
\n" ); document.write( "This is a solution for q. But we want a solution for x. So we substitute back in for q:
\n" ); document.write( " $\"x%5E2+=+%28-sqrt%283%29+%2B+sqrt%2815%29%29%2F2\"$ or $\"x%5E2+=+%28-sqrt%283%29+-+sqrt%2815%29%29%2F2\"$
\n" ); document.write( "Looking at the second equation above we see that the fraction is negative. But a negative numbers cannot be equal to something squared, like $\"x%5E2\"$ ! So unless you are looking for complex solutions, too, there are no solutions to the second equation. We are left with just the first equation:
\n" ); document.write( " $\"x%5E2+=+%28-sqrt%283%29+%2B+sqrt%2815%29%29%2F2\"$
\n" ); document.write( "To solve for x we just find the square root of each side:
\n" ); document.write( " $\"sqrt%28x%5E2%29+=+sqrt%28%28-sqrt%283%29+%2B+sqrt%2815%29%29%2F2%29\"$
\n" ); document.write( " $\"abs%28x%29+=+sqrt%28%28-sqrt%283%29+%2B+sqrt%2815%29%29%2F2%29\"$
\n" ); document.write( " $\"x+=+0+%2B-+sqrt%28%28-sqrt%283%29+%2B+sqrt%2815%29%29%2F2%29\"$
\n" ); document.write( "(Note: The zero is unnecessary mathematically. But Algebra.com's software will not let me use the \"plus or minus\" symbol without a number in front of it.)
\n" ); document.write( "In long form:
\n" ); document.write( " $\"x+=+sqrt%28%28-sqrt%283%29+%2B+sqrt%2815%29%29%2F2%29\"$ or $\"x+=+-sqrt%28%28-sqrt%283%29+%2B+sqrt%2815%29%29%2F2%29\"$

\n" ); document.write( "Eventually, after doing several of these \"quadratic form\"-type equations, you will be able to do these problems without a temporary variable. IOW, you will be able to straight from
\n" ); document.write( " $\"x%5E4+%2B+sqrt%283%29+%2A+x%5E2+-+3+=+0\"$
\n" ); document.write( "to
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