document.write( "Question 251041: how do you create a quadratic equation with only one solution? For example -3? \n" ); document.write( "
Algebra.Com's Answer #182798 by solver91311(24713)\"\" \"About 
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\n" ); document.write( "\n" ); document.write( "Given no other information than rational number solution, such as your example of -3, you can't create a unique quadratic equation. However, if you have either a real irrational solution or a complex solution, then you only have to realize that such solutions always come in conjugate pairs.\r
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\n" ); document.write( "\n" ); document.write( "Let and be any rational numbers. Let be a non-transcendental irrational number. Then if is a root of the quadratic, then is also a solution of the quadratic.\r
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\n" ); document.write( "\n" ); document.write( "Let and be any real numbers. Let be the imaginary number defined by . Then if is a root of the quadratic, then is also a solution of the quadratic.\r
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\n" ); document.write( "\n" ); document.write( "On the other hand, if you know another bit of information about the quadratic for which you have a single rational number solution, then you can derive an equation for the quadratic. For example, we know that the graph of a quadratic of the form is a parabola symmetric to the vertical line that passes through its vertex. So if the vertex of the parabola is at , and the quadratic has real number roots, then the two roots must be equidistant from the vertex, and therefore equidistant from the intersection of the parabola's axis and the -axis, the line . Using your example of -3 for a root, the distance from the intersection of the parabola axis and the -axis is exactly . So, depending on which side of the vertex -3 exists, you either add or subtract the distance value you just calculated from to find the other root.\r
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