1. Start with the Given Equation: We have the quadratic equation x² + 5x + 6 = 0.
2. Move the Constant to the Right Side: We want to isolate the quadratic expression, so we move the constant to the right side: x² + 5x = -6.
3. Add and Subtract (5 / 2)²: To complete the square, we need to find the value that makes x² + 5x a perfect square trinomial. We add and subtract (5 / 2)² inside the equation: x² + 5x + (5 / 2)² – (5 / 2)² = -6.
4. Express as a Perfect Square: We can now express the left side as a perfect square: (x + 5 / 2)² – 25 / 4 = -6.
5. Move the Subtracted Term to the Right Side: We add 25 / 4 to both sides to isolate the perfect square: (x + 5 / 2)² = 1 / 4.
6. Take the Square Root of Both Sides: To solve for x, we take the square root of both sides: x + 5 / 2 = ± 1 / 2.
7. Subtract 5 / 2 from Both Sides: We subtract 5 / 2 from both sides to find the solutions for x: x = -5 / 2 ± 1 / 2.
8. Solve for x (First Solution): The first solution is x = -5 / 2 + 1 / 2 = -2.
9. Solve for x (Second Solution): The second solution is x = -5 / 2 – 1 / 2 = -3.
The solutions to the equation x² + 5x + 6 = 0 are x = -2 and x = -3. This method of solving by expressing the equation in the form (x + a)² = b and using the property of squaring provides a clear and systematic approach to finding the roots of the quadratic equation.