document.write( "Question 766741: show that x^2+(3k-2)x+k(k-1)=0 has real roots for all value of k \n" ); document.write( "

Algebra.Com's Answer #467192 by fcabanski(1391) $\"\"$ $\"About$

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A quadratic equation is $\"ax%5E2+%2B+bx+%2B+c\"$ . Remember that a quadratic equation has real roots if the discriminant is greater than or equal to 0. The discriminant is $\"b%5E2+-+4ac\"$ .

\n" ); document.write( "For the given equation b = 3k - 2 and $\"b%5E2+=+9k%5E2+-+12k+%2B4\"$

\n" ); document.write( "a = 1 and c = k(k-1) = $\"k%5E2+-+k\"$

\n" ); document.write( "The discriminant is therefore (((9k^2 - 12k +4 - 4k^2 +4k = 5k^2 -8k +4}}}

\n" ); document.write( "That is a quadratic equation with a=5, b = -8 and c =4. It has to be greater than or equal to 0 in order for the original equation to have real roots for all values of k.

\n" ); document.write( "The discriminant of this equation is 64-80 = -16. That means this equation has no real roots. In other words, it's a parabola that never crosses the x-axis (it has no real zeroes.) Since a is positive, we know the parabola opens up. Since it opens up, and it never crosses the x-axis, its values are all positive.

\n" ); document.write( " Since the discriminant of the original equation is itself a parabola with only positive values, the original equation has a positive discriminant, and thus only real roots. \n" ); document.write( "

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