Solve by Factoring x^3+125=0

Problem

x3+125=0

Solution

1. Identify the expression on the left side as a sum of two cubes, since 125=5

2. Apply the formula for the sum of cubes, which is a3+b3=(a+b)*(a2−a*b+b2) where a=x and b=5

3. Factor the equation into a linear factor and a quadratic factor.

x3+5=(x+5)*(x2−5*x+25)

4. Set each factor to zero to find the roots of the equation.

(x+5)*(x2−5*x+25)=0

5. Solve the linear factor by setting x+5=0

x=−5

6. Solve the quadratic factor by setting x2−5*x+25=0 and using the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a)

x=(5±√(,(−5)2−4*(1)*(25)))/(2*(1))

7. Simplify the discriminant and the resulting complex roots.

x=(5±√(,25−100))/2

x=(5±√(,−75))/2

x=(5±5*i√(,3))/2

Final Answer

x=−5,(5±5*i√(,3))/2

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