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Solving the Quadratic Equation: (x+7)² - 11 = 0
This article will guide you through the process of solving the quadratic equation (x+7)² - 11 = 0.
Understanding the Equation
The equation (x+7)² - 11 = 0 is a quadratic equation because the highest power of the variable x is 2. To solve it, we'll use the following steps:
1. Expanding the Equation
First, we need to expand the squared term:
(x+7)² = (x+7)(x+7) = x² + 14x + 49
Substituting this back into the original equation:
x² + 14x + 49 - 11 = 0
Simplifying:
x² + 14x + 38 = 0
2. Solving using the Quadratic Formula
The quadratic formula is a general solution for equations in the form ax² + bx + c = 0. In our case, a = 1, b = 14, and c = 38.
The quadratic formula is:
x = (-b ± √(b² - 4ac)) / 2a
Substituting the values:
x = (-14 ± √(14² - 4 * 1 * 38)) / 2 * 1
x = (-14 ± √(196 - 152)) / 2
x = (-14 ± √44) / 2
x = (-14 ± 2√11) / 2
3. Finding the Solutions
We now have two possible solutions:
x1 = (-14 + 2√11) / 2 = -7 + √11
x2 = (-14 - 2√11) / 2 = -7 - √11
Therefore, the solutions to the equation (x+7)² - 11 = 0 are x = -7 + √11 and x = -7 - √11.
Conclusion
We have successfully solved the quadratic equation (x+7)² - 11 = 0 by expanding the equation, applying the quadratic formula, and finding the two possible solutions. This process demonstrates how to solve similar equations by utilizing the quadratic formula and algebraic manipulation.