Solve $x^2+12 x=-11$ by completing the square. Which is the solution set of the equation?
The objective is to solve the quadratic equation \(x^2 + 12x = -11\) using the technique of completing the square. A quadratic equation in the form \(x^2 + bx = c\) can be transformed into a perfect square by adding the square of half the coefficient of \(x\), which is \((\frac{b}{2})^2\), to both sides of the equation. In our case, \(b = 12\), so we calculate \((\frac{12}{2})^2 = 6^2 = 36\). This will allow us to rewrite the quadratic expression as a square of a binomial, making it much easier to isolate \(x\).
Starting with the given equation \(x^2 + 12x = -11\), we add 36 to both sides to complete the square on the left side:
\[x^2 + 12x + 36 = -11 + 36\]
Simplifying the right side, we get:
\[x^2 + 12x + 36 = 25\]
The left side is now a perfect square trinomial, which can be factored as:
\[(x + 6)^2 = 25\]
To solve for \(x\), we take the square root of both sides of the equation \((x + 6)^2 = 25\). This gives us two possible cases for the expression inside the square:
\[x + 6 = 5 \quad \text{or} \quad x + 6 = -5\]
Solving each linear equation separately:
For \(x + 6 = 5\), we subtract 6 from both sides to find \(x = 5 - 6 = -1\).
For \(x + 6 = -5\), we subtract 6 from both sides to find \(x = -5 - 6 = -11\).
Thus, the solutions to the equation are \(-1\) and \(-11\). The solution set is \(\{-11, -1\}\).
Completing the square is a powerful algebraic tool used in various fields, such as physics, engineering, and geometry. For instance, when analyzing the motion of an object under constant acceleration, the path is often described by a quadratic equation. By completing the square, an engineer can easily determine the vertex of the path, which represents the maximum or minimum height of the object. This method helps simplify complex equations, making it easier to understand the behavior of systems in real-world applications.