document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=818956>Question 818956</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The question asks to find a quadratic equation whose solutions are +2i and -2i. How do you find this with no other information? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #492885 by </B><A HREF=/tutors/aboutme.mpl?userid=TimothyLamb><B>TimothyLamb(4379)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=TimothyLamb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=TimothyLamb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=492885\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ---<BR>\n" );
document.write( "the standard form of a quadratic equation is:<BR>\n" );
document.write( "f(x) = ax^2 + bx + c<BR>\n" );
document.write( "---<BR>\n" );
document.write( "for this problem, the roots of the quadratic are given as a complex conjugate-pair:<BR>\n" );
document.write( "+2i and -2i<BR>\n" );
document.write( "---<BR>\n" );
document.write( "the fact that the roots are complex means the discriminant of the quadratic is negative.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "for details on the discriminant read the help text for \"Discriminant\" here:<BR>\n" );
document.write( "https://sooeet.com/math/quadratic-formula-solver.php<BR>\n" );
document.write( "---<BR>\n" );
document.write( "the roots may be calculated from the standard form given above, by using the quadratic formula.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "for details on the quadratic formula read the help text for \"First quadratic root\" here:<BR>\n" );
document.write( "https://sooeet.com/math/quadratic-formula-solver.php<BR>\n" );
document.write( "---<BR>\n" );
document.write( "the fact that the complex roots do not have a real part <BR>\n" );
document.write( "(i.e. the complex roots only have an imaginary part)<BR>\n" );
document.write( "tells us (from the quadratic formula), that b=0<BR>\n" );
document.write( "---<BR>\n" );
document.write( "to see why this is so, read the help text given just above<BR>\n" );
document.write( "---<BR>\n" );
document.write( "so now we can simplify the standard form to:<BR>\n" );
document.write( "f(x) = ax^2 + (0)x + c<BR>\n" );
document.write( "f(x) = ax^2 + c<BR>\n" );
document.write( "---<BR>\n" );
document.write( "the discriminant of the quadratic is: b^2 - 4ac<BR>\n" );
document.write( "but we now know that for this quadratic b=0<BR>\n" );
document.write( "so the discriminant can be simplified to: -4ac<BR>\n" );
document.write( "---<BR>\n" );
document.write( "so for this particular problem, all that remains of the quadratic formula is this:<BR>\n" );
document.write( "x = (+)or(-) sqrt( -4ac )/2a<BR>\n" );
document.write( "---<BR>\n" );
document.write( "and we know from the given roots that x= +2i and x= -2i <BR>\n" );
document.write( "---<BR>\n" );
document.write( "so -4ac < 0 because that is where i comes from: i= sqrt(-1)<BR>\n" );
document.write( "---<BR>\n" );
document.write( "given all of the above, solve for \"a\" in the simplified quadratic formula above:<BR>\n" );
document.write( "---<BR>\n" );
document.write( "x = (+)or(-) sqrt( -4ac )/2a<BR>\n" );
document.write( "x = (+)or(-) sqrt(-1) * sqrt(4ac)/2a<BR>\n" );
document.write( "x = (+)or(-) i * sqrt(4ac)/2a<BR>\n" );
document.write( "---<BR>\n" );
document.write( "use one of the given roots with the simplified quadratic formula above (either root will work, so x = +2i):<BR>\n" );
document.write( "x = +2i = +i * sqrt(4ac)/2a<BR>\n" );
document.write( "---<BR>\n" );
document.write( "sqrt(4ac)/2a = 2<BR>\n" );
document.write( "sqrt(4ac) = 4a<BR>\n" );
document.write( "4ac = 4a4a<BR>\n" );
document.write( "c = 4a<BR>\n" );
document.write( "a = c/4<BR>\n" );
document.write( "---<BR>\n" );
document.write( "answer:<BR>\n" );
document.write( "there is an infinite set of quadratic equations of the form f(x) = ax^2 + bx + c <BR>\n" );
document.write( "where the roots of each equation are x= +2i and x= -2i<BR>\n" );
document.write( "and where:<BR>\n" );
document.write( "a = c/4<BR>\n" );
document.write( "b = 0<BR>\n" );
document.write( "---<BR>\n" );
document.write( "one such equation is this: <BR>\n" );
document.write( "f(x) = x^2 + 4<BR>\n" );
document.write( "---<BR>\n" );
document.write( "check:<BR>\n" );
document.write( "the above quadratic equation is in standard form, with a=1, b=0, and c=4<BR>\n" );
document.write( "---<BR>\n" );
document.write( "to solve the quadratic equation by using the quadratic formula, plug this:<BR>\n" );
document.write( "1 0 4<BR>\n" );
document.write( "into this: https://sooeet.com/math/quadratic-equation-solver.php<BR>\n" );
document.write( "---<BR>\n" );
document.write( "this quadratic has complex roots:<BR>\n" );
document.write( "---<BR>\n" );
document.write( "the two complex roots of the quadratic are:<BR>\n" );
document.write( "x = 0 + (2)i<BR>\n" );
document.write( "x = 0 - (2)i<BR>\n" );
document.write( "---<BR>\n" );
document.write( "these are the same roots given in the problem statement, so our result is correct !<BR>\n" );
document.write( "---<BR>\n" );
document.write( "Solve and graph linear equations:<BR>\n" );
document.write( "https://sooeet.com/math/linear-equation-solver.php<BR>\n" );
document.write( "---<BR>\n" );
document.write( "Solve quadratic equations, quadratic formula:<BR>\n" );
document.write( "https://sooeet.com/math/quadratic-formula-solver.php<BR>\n" );
document.write( "---<BR>\n" );
document.write( "Calculate and graph the linear regression of any data set:<BR>\n" );
document.write( "https://sooeet.com/math/linear-regression.php </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );