- Identify the coefficient of the \(x\)-term as \(6\), divide by \(2\) to get \(3\), and square it to get \(9\).
- Add \(9\) to both sides of the equation \(x^2 + 6x = 7\) to get \(x^2 + 6x + 9 = 16\).
- Factor the left side to get \((x+3)^2 = 16\).
- Solve for \(x\) by taking the square root to get \(x+3 = \pm 4\), resulting in the solution set \(\{-7, 1\}\).
- The solution set is \(\boxed{\{-7, 1\}}\).
Explanation
- Problem Analysis
The objective is to solve the quadratic equation \(x^2 + 6x = 7\) by the method of completing the square. A quadratic equation in the form \(x^2 + bx = c\) can be solved by adding the square of half the coefficient of the \(x\)-term to both sides of the equation. This allows the left side to be written as a perfect square trinomial of the form \((x + d)^2\).
- Completing the Square
We begin with the equation \(x^2 + 6x = 7\). To complete the square, we identify the coefficient of the \(x\)-term, which is \(6\). We divide this coefficient by \(2\) to get \(3\), and then square it to get \(3^2 = 9\). We add this value, \(9\), to both sides of the equation to maintain equality:
\[x^2 + 6x + 9 = 7 + 9\]
\[x^2 + 6x + 9 = 16\]
- Solving for x
Now that we have added \(9\) to both sides, the left side, \(x^2 + 6x + 9\), is a perfect square. We can factor it as \((x + 3)^2\). So the equation becomes:
\[(x + 3)^2 = 16\]
Taking the square root of both sides gives us two possible cases for the expression inside the square:
\[x + 3 = 4 \quad \text{or} \quad x + 3 = -4\]
- Conclusion
Now, we solve each case for \(x\):
For the first case: \(x + 3 = 4 \implies x = 4 - 3 = 1\).
For the second case: \(x + 3 = -4 \implies x = -4 - 3 = -7\).
Thus, the solutions to the equation are \(x = 1\) and \(x = -7\). Comparing this to the given options, the correct solution set is \(\{-7, 1\}\).
Examples
Completing the square is a fundamental technique in algebra that is often used to derive the quadratic formula or to transform quadratic functions into vertex form. For instance, if you are designing a path for a projectile, such as a ball thrown into the air, the equation for its height over time is often a quadratic. By completing the square, you can easily find the maximum height the ball reaches and the time it takes to reach it. This skill is vital for solving real-world optimization problems where you need to identify the minimum or maximum value of a quadratic relationship.