2020 AMC 10A Problems/Problem 5
What is the sum of all real numbers for which
Solution 1 (Casework and Factoring)
Split the equation into two cases, where the value inside the absolute value is positive and nonpositive.
The equation yields , which is equal to . Therefore, the two values for the positive case is and .
Similarly, taking the nonpositive case for the value inside the absolute value notation yields . Factoring and simplifying gives , so the only value for this case is .
Summing all the values results in .
Solution 2 (Casework and Vieta)
We have the equations and .
Notice that the second is a perfect square with a double root at , and the first has two distinct real roots. By Vieta's, the sum of the roots of the first equation is or . .
Solution 3 (Casework and Graphing)
Completing the square gives Note that the graph of is an upward parabola with the vertex and the axis of symmetry the graphs of are horizontal lines.
We apply casework to
The line intersects the parabola at two points. Since these two points are symmetric about the line the average of their -coordinates is
In this case, the average of the solutions is so the sum of the solutions is
The line intersects the parabola at one point, which is the vertex.
In this case, the only solution is
Finally, the sum of all solutions is
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