Asked by Nico
Solve x^3 = 125
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GPT-5 mini
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Solve x^3 = 125.
Since 125 = 5^3, one real solution is x = 5.
Factor x^3 − 125 = (x − 5)(x^2 + 5x + 25). Solve the quadratic:
x = [−5 ± sqrt(25 − 100)]/2 = [−5 ± i·5√3]/2 = −5/2 ± i(5√3/2).
So the three cube roots are:
- x = 5
- x = −5/2 + i(5√3/2)
- x = −5/2 − i(5√3/2)
(Or x = 5·e^{2πik/3} for k = 0,1,2.)
Since 125 = 5^3, one real solution is x = 5.
Factor x^3 − 125 = (x − 5)(x^2 + 5x + 25). Solve the quadratic:
x = [−5 ± sqrt(25 − 100)]/2 = [−5 ± i·5√3]/2 = −5/2 ± i(5√3/2).
So the three cube roots are:
- x = 5
- x = −5/2 + i(5√3/2)
- x = −5/2 − i(5√3/2)
(Or x = 5·e^{2πik/3} for k = 0,1,2.)
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