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To get two equal, real roots, we need a perfect square trinomial, or an equation that can be factored into something of the form (ax+b)^2=0, as opposed to (ax+b)(cx+d)=0. As you might have learned, (ax+b)^2=0 expands to (a^2)(x^2)+2ab+b^2=0. So the equation you were given, 4x^2+(k+7)x+(k+4)=0, will change to (2x+___)^2=0.
\n" ); document.write( "The tricky part is dealing with all of these k terms. We have 4x^2+(k+7)x+(k+4)=0 and we want (a^2)(x^2)+2ab+b^2=0. That means b^2=k+4, so b must be sqrt(k+4). a^2=4, so a=2. And finally, 2ab=k+7, for the middle term.\r
\n" ); document.write( "\n" ); document.write( "So:
\n" ); document.write( "a=2
\n" ); document.write( "b=sqrt(k+4)
\n" ); document.write( "2ab=k+7
\n" ); document.write( "So, using the first 2 equations in the third, 2(2)(sqrt(k+4))=k+7
\n" ); document.write( "4sqrt(k+4)=k+7
\n" ); document.write( "Square all of these terms to get rid of the sqrt
\n" ); document.write( "16(k+4)=(k+7)^2
\n" ); document.write( "16k+64=k^2+14k+49
\n" ); document.write( "Get all of the terms to the right side (because I want k^2 to stay k^2, and not to become -k^2)
\n" ); document.write( "0=k^2-2k-15
\n" ); document.write( "Factor the expression on the right
\n" ); document.write( "0=(k-5)(k+3)
\n" ); document.write( "k=5,-3
\n" ); document.write( "You can test these values in your original equation, and you should get two perfect square trinomials. Good luck!
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