Asked by bob
𝑥^3 −9𝑥^2 +27𝑥 =20 solve by graphing
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GPT-5 mini
AI
Rewrite the equation as
f(x) = x^3 − 9x^2 + 27x − 20 = 0.
Notice x^3 − 9x^2 + 27x = (x − 3)^3 + 27, so
(x − 3)^3 + 27 = 20 ⇒ (x − 3)^3 = −7.
Taking cube roots gives the unique real solution
x = 3 − 7^(1/3) ≈ 3 − 1.912931 ≈ 1.08707.
(You can see this by graphing y1 = x^3 − 9x^2 + 27x and y2 = 20 — they intersect once at x ≈ 1.08707. The derivative f'(x)=3(x−3)^2 ≥ 0, so the cubic is monotone and has exactly one real root.)
f(x) = x^3 − 9x^2 + 27x − 20 = 0.
Notice x^3 − 9x^2 + 27x = (x − 3)^3 + 27, so
(x − 3)^3 + 27 = 20 ⇒ (x − 3)^3 = −7.
Taking cube roots gives the unique real solution
x = 3 − 7^(1/3) ≈ 3 − 1.912931 ≈ 1.08707.
(You can see this by graphing y1 = x^3 − 9x^2 + 27x and y2 = 20 — they intersect once at x ≈ 1.08707. The derivative f'(x)=3(x−3)^2 ≥ 0, so the cubic is monotone and has exactly one real root.)
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