document.write( "Question 1172144: For this problem: All the roots of \n" );
document.write( "x^2 + px + q = 0 are real, where p and q are real numbers. Prove that all the roots of \n" );
document.write( "x^2 + px + q + (x + a)(2x + p) = 0 are real, for any real number a. \n" );
document.write( "I solved it like this: \n" );
document.write( "We know that all roots of x^2+px+q=0 are real. We can derive from this condition that p^2- 4q >=0 \n" );
document.write( "Let us do some simplification of second equation: \n" );
document.write( "x^2+ px + q (x+a)(2x+p) = 3x^2+ x(2p + 2a) + ap + q \n" );
document.write( "So we want to prove that the discriminant of equation \n" );
document.write( "3x^2+x(2p + 2a) + ap + q = 0 \n" );
document.write( "is greater or equal to zero. \n" );
document.write( "D = (2p + 2a)^2 - 4 *3(ap+q) = 4(p^2 + a^2 + 2ap - 3ap - 3q) = 4(a^2-ap+p^2-3q) \n" );
document.write( "To prove that D >=0 , we can view D as a polynomial of a: \n" );
document.write( "D(a) = 4a^2 - a(4p) + 4(p^2-3q) \n" );
document.write( "We know that if a quadratic polynomial has a positive greatest coefficient and it's discriminant <= 0 then polynomials are always positive. \n" );
document.write( "So it remains for us to prove that \n" );
document.write( "(4p)^2 - 4*4*4(p^2-3q) <= 0 \n" );
document.write( "Let's divide both side by 4^2 : \n" );
document.write( "p^2 - 4(p^2-3q) <= 0 \n" );
document.write( "-3p^2 + 12q <= 0 \n" );
document.write( "Now we divide by 3 \n" );
document.write( "-p^2+4q<= 0 \n" );
document.write( "Now we transfer terms to the other side: \n" );
document.write( "0<= p^2 -4q, or p^2-4q >=0 \n" );
document.write( "For this problem, is there any other ways to solve it? If so, can you show me? Thanks! \n" );
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