document.write( "Question 1209598: What is the domain of the function f(x) = sqrt(6x - 3 + x^2)? \n" ); document.write( "
Algebra.Com's Answer #849672 by CPhill(1959)\"\" \"About 
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The domain of a square root function is restricted to non-negative values under the radical. Therefore, we need to find the values of x for which:\r
\n" ); document.write( "\n" ); document.write( "6x - 3 + x² ≥ 0\r
\n" ); document.write( "\n" ); document.write( "Rearranging the terms, we get:\r
\n" ); document.write( "\n" ); document.write( "x² + 6x - 3 ≥ 0\r
\n" ); document.write( "\n" ); document.write( "To find the values of x that satisfy this inequality, we first find the roots of the corresponding quadratic equation:\r
\n" ); document.write( "\n" ); document.write( "x² + 6x - 3 = 0\r
\n" ); document.write( "\n" ); document.write( "Using the quadratic formula:\r
\n" ); document.write( "\n" ); document.write( "x = (-b ± √(b² - 4ac)) / 2a
\n" ); document.write( "x = (-6 ± √(6² - 4 * 1 * -3)) / 2 * 1
\n" ); document.write( "x = (-6 ± √(36 + 12)) / 2
\n" ); document.write( "x = (-6 ± √48) / 2
\n" ); document.write( "x = (-6 ± 4√3) / 2
\n" ); document.write( "x = -3 ± 2√3\r
\n" ); document.write( "\n" ); document.write( "So, the roots are x = -3 - 2√3 and x = -3 + 2√3.\r
\n" ); document.write( "\n" ); document.write( "Since the quadratic has a positive leading coefficient, the parabola opens upwards. This means the inequality x² + 6x - 3 ≥ 0 is satisfied when x is less than or equal to the smaller root, or x is greater than or equal to the larger root.\r
\n" ); document.write( "\n" ); document.write( "Therefore, the domain of f(x) is:\r
\n" ); document.write( "\n" ); document.write( "(-∞, -3 - 2√3] ∪ [-3 + 2√3, ∞)
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