Key to "If x=5−523−513, prove that x3−15x2+60x−20=0?"

Alaina Ortiz

Answered question

2022-02-06

x = 5 - 5^{\frac{2}{3}} - 5^{\frac{1}{3}}

, prove that

x^{3} - 15 x^{2} + 60 x - 20 = 0

tacalaohn

Beginner2022-02-07Added 13 answers

Step 1

x = 5 - 5^{\frac{2}{3}} - 5^{\frac{1}{3}}

5 - x = (5^{\frac{2}{3}} + 5^{\frac{1}{3}})

....(*)

{(5 - x)}^{3} = {(5^{\frac{2}{3}} + 5^{\frac{1}{3}})}^{3}

5^{3} - x^{3} - 3 (5) (x) (5 - x) = {(5^{\frac{2}{3}})}^{3} + {(5^{\frac{1}{3}})}^{3}

+ 3 \times 5^{\frac{2}{3}} \times 5^{\frac{1}{3}} (5^{\frac{2}{3}} + 5^{\frac{1}{3}})

, i.e.,

125 - x^{3} - 75 x + 15 x^{2} = 5^{2} + 5^{1} + 3 \times 5 \times (5 - x) \dots [∴, (\cdot)]

125 - 75 x + 15 x^{2} - x^{3} = 30 + 75 - 15 x

20 - 60 x + 15 x^{2} - x^{3} = 0

, as desired!

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