Solving Square Root Equations

To determine the value of x in the given algebraic equation, we will perform a series of standard mathematical operations. The equation provided is:

\(\sqrt {1 + \frac{x}{{225}}} = \frac{{17}}{{15}}\)

Equation Simplification

Our primary goal is to isolate x. The presence of a square root requires us to eliminate it first, which can be done by squaring both sides of the equation. Following that, we will manage the fractions to solve for x.

Step-by-Step Calculation for X

1. Square Both Sides: To remove the square root from the left side of the equation, we square both sides. \[{\left(\sqrt {1 + \frac{x}{{225}}} \right)^2} = {\left( {\frac{{17}}{{15}}} \right)^2}\] This operation simplifies the equation to: \[1 + \frac{x}{{225}} = \frac{{{17^2}}}{{{15^2}}}\]
2. Evaluate Squares: Now, calculate the square of the numerator and the denominator on the right-hand side. \[17^2 = 289\] \[15^2 = 225\] Substituting these values, the equation becomes: \[1 + \frac{x}{{225}} = \frac{{289}}{{225}}\]
3. Isolate the X Term: To get the term containing x by itself, subtract 1 from both sides of the equation. \[\frac{x}{{225}} = \frac{{289}}{{225}} - 1\] To perform the subtraction, express 1 with the same denominator as the fraction: \(1 = \frac{{225}}{{225}}\). \[\frac{x}{{225}} = \frac{{289}}{{225}} - \frac{{225}}{{225}}\] Subtracting the numerators: \[\frac{x}{{225}} = \frac{{289 - 225}}{{225}}\] \[\frac{x}{{225}} = \frac{{64}}{{225}}\]
4. Determine Value of X: Since the denominators on both sides of the equation are identical (225), the numerators must also be equal for the equation to hold true. Alternatively, multiply both sides by 225. \[225 \times \frac{x}{{225}} = 225 \times \frac{{64}}{{225}}\] This leads directly to the value of x: \[x = 64\]

Conclusion for X

Based on the step-by-step algebraic manipulation, the value of x that satisfies the given equation \(\sqrt {1 + \frac{x}{{225}}} = \frac{{17}}{{15}}\) is 64.