Question 251041: how do you create a quadratic equation with only one solution? For example -3?
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
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.
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.
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.
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 \ =\ y\ =\ ax^2\ +\ bx\ +\ c) is a parabola symmetric to the vertical line that passes through its vertex. So if the vertex of the parabola is at \right)) , 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.
John
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