2-4%3D0.jpeg "(x+7)2-4=0")
Solving the Quadratic Equation: (x+7)² - 4 = 0
This article will guide you through the process of solving the quadratic equation (x+7)² - 4 = 0. We will use various methods to find the solutions, providing a comprehensive understanding of the problem.
1. Expanding and Simplifying
First, we can expand the squared term:
(x+7)² = (x+7)(x+7) = x² + 14x + 49
Now, substitute this back into the original equation:
x² + 14x + 49 - 4 = 0
Simplifying the equation gives us:
x² + 14x + 45 = 0
2. Factoring
We can now factor the quadratic expression:
(x + 5)(x + 9) = 0
This means either:
x + 5 = 0 or x + 9 = 0
Solving for x in each case, we get:
x = -5 or x = -9
3. Quadratic Formula
Alternatively, we can use the quadratic formula to solve for x:
x = (-b ± √(b² - 4ac)) / 2a
where a = 1, b = 14, and c = 45 from the simplified equation x² + 14x + 45 = 0.
Substituting these values into the formula:
x = (-14 ± √(14² - 4 * 1 * 45)) / 2 * 1
x = (-14 ± √(196 - 180)) / 2
x = (-14 ± √16) / 2
x = (-14 ± 4) / 2
This gives us two solutions:
x = (-14 + 4) / 2 = -5
x = (-14 - 4) / 2 = -9
4. Conclusion
Therefore, the solutions to the quadratic equation (x+7)² - 4 = 0 are x = -5 and x = -9. We have solved this equation using multiple methods, demonstrating the versatility and power of different approaches in solving quadratic equations.