To solve the equation \(\frac{(x+16)^2}{9} = 7\), we start by eliminating the fraction by multiplying both sides by 9:
\[ (x + 16)^2 = 63 \]
Next, we take the square root of both sides:
\[ x + 16 = \pm \sqrt{63} \]
Calculating \(\sqrt{63}\):
\[ \sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7} \]
We can also calculate \(3\sqrt{7}\) numerically. The approximate value of \(\sqrt{7}\) is about 2.6457513110645906, so:
\[ 3\sqrt{7} \approx 3 \cdot 2.6457513110645906 \approx 7.937 \]
Now substituting back into our equation gives us:
\[ x + 16 = \pm 7.937 \]
This results in two cases:
- \(x + 16 = 7.937\)
- \(x + 16 = -7.937\)
Case 1:
\[ x + 16 = 7.937 \implies x = 7.937 - 16 \implies x \approx -8.063 \]
Case 2:
\[ x + 16 = -7.937 \implies x = -7.937 - 16 \implies x \approx -23.937 \]
Thus, we can summarize the solutions as:
\[ x \approx -8.06 \quad \text{and} \quad x \approx -23.94 \]
Finally, rounding to the nearest hundredth, the solutions are:
\[ \boxed{-8.06} \quad \text{and} \quad \boxed{-23.94} \]
1 answer