To solve the quadratic equation \((x + 27)^2 - 6 = -3\), we'll follow these steps:
- Isolate the squared term: Start by adding 6 to both sides of the equation.
\[ (x + 27)^2 - 6 + 6 = -3 + 6 \] \[ (x + 27)^2 = 3 \]
- Take the square root of both sides: When taking the square root, remember to include both the positive and negative roots.
\[ x + 27 = \sqrt{3} \quad \text{or} \quad x + 27 = -\sqrt{3} \]
- Solve for \(x\): Subtract 27 from both sides for both cases.
For \(x + 27 = \sqrt{3}\):
\[ x = \sqrt{3} - 27 \]
For \(x + 27 = -\sqrt{3}\):
\[ x = -\sqrt{3} - 27 \]
- Calculate the numerical values: Using the approximation \(\sqrt{3} \approx 1.732\):
- For \(x = \sqrt{3} - 27\):
\[ x \approx 1.732 - 27 \approx -25.268 \]
- For \(x = -\sqrt{3} - 27\):
\[ x \approx -1.732 - 27 \approx -28.732 \]
- Round the results to the nearest hundredth:
-
First solution: \(x \approx -25.27\)
-
Second solution: \(x \approx -28.73\)
Thus, the solutions to the quadratic equation are:
\[ x \approx -25.27 \quad \text{and} \quad x \approx -28.73 \]
1 answer