Find the value of ( x ) if ( cos(2x) = frac{1}{2} ) and ( x^3 + 1 = 0 )

Question

Accepted Answer

Step:1 ：First we solve the equation ( cos(2x) = frac{1}{2} ). The general solution to this equation is ( x = frac{pi}{3} + 2npi ) or ( x = -frac{pi}{3} + 2npi ), where ( n ) is an integer.Step:2 ：Next, we solve the equation ( x^3 + 1 = 0 ). This is a sum of cubes equation. It can be factored into ( (x+1)(x^2 - x + 1) = 0 ). Setting each factor equal to zero gives us ( x = -1 ) or ( x^2 - x + 1 = 0 ). The second equation has no real solutions, so ( x = -1 ) is the only solution to this equation.Step:3 ：The final step is to find the common solutions to both equations. The only common solution is ( x = -1 ), which corresponds to ( n = -1 ) in the first equation.

Answer

Step: 1 ：First we solve the equation ( cos(2x) = frac{1}{2} ). The general solution to this equation is ( x = frac{pi}{3} + 2npi ) or ( x = -frac{pi}{3} + 2npi ), where ( n ) is an integer.Step: 2 ：Next, we solve the equation ( x^3 + 1 = 0 ). This is a sum of cubes equation. It can be factored into ( (x+1)(x^2 - x + 1) = 0 ). Setting each factor equal to zero gives us ( x = -1 ) or ( x^2 - x + 1 = 0 ). The second equation has no real solutions, so ( x = -1 ) is the only solution to this equation.Step: 3 ：The final step is to find the common solutions to both equations. The only common solution is ( x = -1 ), which corresponds to ( n = -1 ) in the first equation.

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