2%E2%88%9211%3D0.jpeg "(x+7)2−11=0")
Solving the Quadratic Equation: (x+7)² - 11 = 0
This article explores the process of solving the quadratic equation (x+7)² - 11 = 0. We'll break down the steps using the most common methods:
1. Expanding and Simplifying
First, we expand the square:
(x+7)² = (x+7)(x+7) = x² + 14x + 49
Now, substitute this back into the original equation:
x² + 14x + 49 - 11 = 0
Simplify:
x² + 14x + 38 = 0
2. Using the Quadratic Formula
The quadratic formula provides a direct solution for any equation in the form ax² + bx + c = 0:
x = (-b ± √(b² - 4ac)) / 2a
In our equation, a = 1, b = 14, and c = 38. Substitute these values into the formula:
x = (-14 ± √(14² - 4 * 1 * 38)) / (2 * 1)
Simplify:
x = (-14 ± √(196 - 152)) / 2
x = (-14 ± √44) / 2
x = (-14 ± 2√11) / 2
Finally, simplify further:
x = -7 ± √11
Therefore, the solutions for the equation (x+7)² - 11 = 0 are:
x = -7 + √11 and x = -7 - √11
3. Completing the Square
This method involves manipulating the equation to create a perfect square trinomial. We start by moving the constant term to the right side:
x² + 14x = 11
To complete the square, we take half of the coefficient of our x term (which is 14), square it (14/2 = 7, 7² = 49), and add it to both sides:
x² + 14x + 49 = 11 + 49
Now we can factor the left side as a perfect square:
(x + 7)² = 60
Take the square root of both sides:
x + 7 = ±√60
Simplify and solve for x:
x = -7 ± 2√15
Therefore, the solutions are:
x = -7 + 2√15 and x = -7 - 2√15
Conclusion
We successfully solved the quadratic equation (x+7)² - 11 = 0 using three different methods: expanding and simplifying, the quadratic formula, and completing the square. Each method yielded the same solutions, demonstrating the versatility of solving quadratic equations.