If p²-3p-1=0,find
iii)p³-1/p³
Answers
Answer:
p² - 3p + 1 = 0
_____________ [GIVEN]
• We have to find the value of p² + \dfrac{1}{ {p}^{2} }
p
2
1
______________________________
Solution:
\implies⟹ p² - 3p + 1 = 0
\implies⟹ p² + 1 = 3p
Divide by p on both sides
\implies⟹ \dfrac{ {p}^{2} }{p}
p
p
2
+ \dfrac{1}{p}
p
1
= \dfrac{3p}{p}
p
3p
\implies⟹ p + \dfrac{1}{p}
p
1
= 3
Now.. squaring on both sides..
\implies⟹ \bigg ({p \: + \: \dfrac{1}{p}} \bigg)^{2}(p+
p
1
)
2
= (3)²
• (a + b)² = a² + b² + 2ab
\implies⟹ \bigg ( {p}^{2} \: + \: \dfrac{1}{ {p}^{2} } \: + \: 2p \: \times \: \dfrac{1}{p} \bigg)(p
2
+
p
2
1
+2p×
p
1
) = 9
\implies⟹ p² + \dfrac{1}{ {p}^{2} }
p
2
1
+ 2 = 9
\implies⟹ p² + \dfrac{1}{ {p}^{2} }
p
2
1
= 9 - 7
_____________________________
\huge{\bold{{p}{^2}\:+\:\dfrac{1}{p}^{2} \: =\:7}}p
2
+
p
1
2
=7
Step-by-step explanation:
p² - 3p + 1 = 0
_____________ [GIVEN]
• We have to find the value of p² + \dfrac{1}{ {p}^{2} }p21
______________________________
Solution:
\implies⟹ p² - 3p + 1 = 0
\implies⟹ p² + 1 = 3p
Divide by p on both sides
\implies⟹ \dfrac{ {p}^{2} }{p}pp2 + \dfrac{1}{p}p1 = \dfrac{3p}{p}p3p
\implies⟹ p + \dfrac{1}{p}p1 = 3
Now.. squaring on both sides..
\implies⟹ \bigg ({p \: + \: \dfrac{1}{p}} \bigg)^{2}(p+p1)2 = (3)²
• (a + b)² = a² + b² + 2ab
\implies⟹ \bigg ( {p}^{2} \: + \: \dfrac{1}{ {p}^{2} } \: + \: 2p \: \times \: \dfrac{1}{p} \bigg)(p2+p21+2p×p1) = 9
\implies⟹ p² + \dfrac{1}{ {p}^{2} }p21 + 2 = 9
\implies⟹ p² + \dfrac{1}{ {p}^{2} }p21 = 9 - 7
_____________________________
\huge{\bold{{p}{^2}\:+\:\dfrac{1}{p}^{2} \: =\:7}}p2+p12=7
_____________ \bold{[ANSWER]}[ANSWER]
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