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√(2x -1) = 2x -7
Step-by-step explanation:
It helps to have a general idea of the shape of the square root curve. The coefficient 'a' will compress it horizontally by that factor, and the constant 'b' will add a left shift.
To make the equation have only one solution, the line must intersect the curve in only one place. That is, its slope must be sufficiently steep if it is positive, or must be negative. The y-intercept must be chosen so the line will intersect the square root curve.
It can work well in this case to decide what the point of intersection will be before you finish writing the equation.
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Suppose we choose a=2 and b=-1 to create a square root curve that starts to the right of the y-axis and opens to the right. (The "left shift" is negative.) For these values, we will get an integer solution when 2x-1 is an odd square. That is, suitable x-values could be 1, 5, 13, 25, and corresponding y-values would be 1, 3, 5, 7.
Any line that passes through any of these points, but does not cross the square root curve again, will do. The one I proposed in the attachment goes through the point (5, 3) and has a slope of 2. (c=2, d=-7)
With these values, the equation becomes ...
√(2x -1) = 2x -7
and has the solution (x, y) = (5, 3). Solved in the conventional way, one would also see the extraneous solution (2.5, -2).
