Full solution

Q. Solve for

x

\newline

\sqrt{(x-5)^{2}}=\sqrt{18}

Recognize Simplification: We start by recognizing that the square root of a square cancels out, so we can simplify $\sqrt{(x-5)^{2}}$ to $|x-5|$ . This is because the square root and square are inverse operations, but we must consider that the square root function outputs the non-negative root, hence the absolute value. $\newline$ Calculation: $\sqrt{(x-5)^{2}} = |x-5|$
Set Absolute Value: Next, we set the absolute value of $x-5$ equal to the square root of $18$ , which gives us two possible equations: $x-5 = \sqrt{18}$ or $x-5 = -\sqrt{18}$ . This is because the absolute value of a number can be either positive or negative. $\newline$ Calculation: $|x-5| = \sqrt{18}$ leads to $x-5 = \sqrt{18}$ or $x-5 = -\sqrt{18}$
Solve for x ( $1$ st Equation): We solve the first equation $x-5 = \sqrt{18}$ . To find x, we add $5$ to both sides of the equation. $\newline$ Calculation: $x = \sqrt{18} + 5$
Solve for x ( $2$ nd Equation): We solve the second equation $x-5 = -\sqrt{18}$ . Similarly, we add $5$ to both sides of this equation as well. $\newline$ Calculation: $x = -\sqrt{18} + 5$
Simplify $\sqrt{18}$ : Now we simplify $\sqrt{18}$ . The prime factorization of $18$ is $2 \times 3^2$ . We can take the square root of $3^2$ out of the square root, which gives us $3$ . $\newline$ Calculation: $\sqrt{18} = \sqrt{2 \times 3^2} = 3 \times \sqrt{2}$
Substitute Simplified Form: We substitute the simplified form of $\sqrt{18}$ back into the equations for $x$ . $\newline$ Calculation: $x = 3 \times \sqrt{2} + 5$ and $x = -3 \times \sqrt{2} + 5$
Final Solutions: We now have two possible solutions for $x$ , which are $x = 3 \times \sqrt{2} + 5$ and $x = -3 \times \sqrt{2} + 5$ .