The number of roots of the equation $x^2 - 2x - log_2 |1 - x| = 3$ is
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4
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2
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1
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0
The equation x^2 - 2x - 3 = log_2 |1 - x| can be solved by graphing or checking intersection points. The function f(x) = x^2 - 2x - 3 is a parabola, and g(x) = log_2 |1 - x| has vertical asymptotes at x = 1. Analyzing the intersections shows 4 distinct roots.
The equation can be rewritten as x squared minus 2 x minus 3 equaling the base 2 logarithm of the absolute value of 1 minus x. Factoring the left side gives the product of x minus 3 and x plus 1 equaling the base 2 logarithm of the absolute value of 1 minus x. By analyzing the graphs of the quadratic parabola and the logarithmic function, they intersect at multiple points. Accounting for intersections in both the domain where x is less than 1 and the domain where x is greater than 1, there are exactly four distinct solutions to the equation.