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If $x +\frac{1}{x} = 8$ , what is $x$

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Find limits using addition, subtraction, and multiplication laws

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Q. If

x +\frac{1}{x} = 8

, what is

x

Identify equation and ask: Identify the equation given and what is asked. $\newline$ We are given the equation $x + \frac{1}{x} = 8$ and we need to find the value of $x$ .
Eliminate fraction by multiplication: Multiply both sides of the equation by $x$ to eliminate the fraction. $\newline$ $(x)(x) + \left(\frac{1}{x}\right)(x) = 8(x)$ $\newline$ This gives us $x^2 + 1 = 8x$ .
Rearrange to form quadratic equation: Rearrange the equation to form a quadratic equation. $\newline$ Bring all terms to one side of the equation to set it equal to zero. $\newline$ $x^2 - 8x + 1 = 0$
Solve using quadratic formula: Solve the quadratic equation using the quadratic formula. $\newline$ The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = 1$ , $b = -8$ , and $c = 1$ .
Calculate discriminant to determine roots: Calculate the discriminant $b^2 - 4ac$ to determine the nature of the roots. $\newline$ Discriminant = $(-8)^2 - 4(1)(1) = 64 - 4 = 60$
Calculate possible values using formula: Calculate the two possible values for $x$ using the quadratic formula. $\newline$ $x = \frac{8 \pm \sqrt{60}}{2}$ $\newline$ $x = \frac{8 \pm 2\sqrt{15}}{2}$ $\newline$ $x = 4 \pm \sqrt{15}$
Check solutions satisfy original equation: Check both solutions to ensure they satisfy the original equation. $\newline$ For $x = 4 + \sqrt{15}$ , check if $(4 + \sqrt{15}) + \frac{1}{(4 + \sqrt{15})} = 8$ . $\newline$ For $x = 4 - \sqrt{15}$ , check if $(4 - \sqrt{15}) + \frac{1}{(4 - \sqrt{15})} = 8$ .

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