Question

If x-1/x=5 find the value of 1.x^2+1/x^2 2.x^4+1/x^4​

Answer 1

$\rm\bullet\quad x-\dfrac{1}{x}=5$

Squaring both the sides,

$\rm\dashrightarrow\Bigg ( x-\dfrac{1}{x}\Bigg )^2=(5)^2$

Since, (a-b)² = a²+b²-2ab, ∴

$\rm\dashrightarrow (x)^2+\Bigg (\dfrac{1}{x}\Bigg )^2 - 2 \times x \times \dfrac{1}{x}=25$

$\rm\dashrightarrow x^2+\dfrac{1}{x^2} - 2=25$

$\rm\dashrightarrow x^2+\dfrac{1}{x^2} = 25+2$

$\bf\bullet\dashrightarrow x^2+\dfrac{1}{x^2} = 27$

Squaring both the sides again,

$\rm\dashrightarrow \Bigg (x^2+\dfrac{1}{x^2}\Bigg )^2 = (27)^2$

Since, (a+b)²=a²+b²+2ab,∴

$\rm\dashrightarrow (x^2)^2+\Bigg ( \dfrac{1}{x^2} \Bigg )^2 + 2 \times x^2 \times \dfrac{1}{x^2} = 729$

$\rm\dashrightarrow x^4 +\dfrac{1}{x^4} + 2 = 729$

$\rm\dashrightarrow x^4 +\dfrac{1}{x^4} = 729-2$

$\bf\bullet\dashrightarrow x^4 +\dfrac{1}{x^4} = 727$