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solve for $x$ : $x=\sqrt{2x-4} +8$

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Evaluate absolute value expressions

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Q. solve for

x

:

x=\sqrt{2x-4} +8

Isolate square root: Step $1$ : Start by isolating the square root on one side of the equation. $\newline$ $x - 8 = \sqrt{2x - 4}$
Square both sides: Step $2$ : Square both sides to eliminate the square root. $\newline$ $(x - 8)^2 = (2x - 4)$
Expand squared term: Step $3$ : Expand the squared term. $x^2 - 16x + 64 = 2x - 4$
Bring terms together: Step $4$ : Bring all terms to one side to set the equation to zero. $\newline$ $x^2 - 18x + 68 = 0$
Use quadratic formula: Step $5$ : Use the quadratic formula to solve for $x$ . The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = 1$ , $b = -18$ , and $c = 68$ . $x = \frac{18 \pm \sqrt{(-18)^2 - 4\cdot1\cdot68}}{2\cdot1}$ $x = \frac{18 \pm \sqrt{324 - 272}}{2}$ $x = \frac{18 \pm \sqrt{52}}{2}$
Simplify square root: Step $6$ : Simplify $\sqrt{52}$ . $\newline$ $\sqrt{52} \approx 7.21$ $\newline$ $x = (18 \pm 7.21) / 2$ $\newline$ $x = (18 + 7.21) / 2$ or $x = (18 - 7.21) / 2$ $\newline$ $x \approx 12.605$ or $x \approx 5.395$
Check solutions: Step $7$ : Check both solutions in the original equation to verify. $\newline$ For $x \approx 12.605$ : $\newline$ $12.605 = \sqrt{(2\cdot12.605 - 4)} + 8$ $\newline$ $12.605 \approx \sqrt{21.21} + 8$ $\newline$ $12.605 \approx 4.605 + 8$ $\newline$ $12.605 \approx 12.605$ (Correct) $\newline$ For $x \approx 5.395$ : $\newline$ $5.395 = \sqrt{(2\cdot5.395 - 4)} + 8$ $\newline$ $5.395 \approx \sqrt{6.79} + 8$ $\newline$ $5.395 \approx 2.605 + 8$ $\newline$ $5.395 \approx 10.605$ (Incorrect)

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