Introduction: Solving Quadratic Equations Using Completing the Square

Quadratic equations are fundamental in mathematics and are commonly encountered in various fields such as physics, engineering, and economics. One method to solve quadratic equations is by completing the square. This technique involves transforming a quadratic equation into a perfect square trinomial, making it easier to find the solution set. By mastering the process of completing the square, you can efficiently solve quadratic equations like x^2 = 12x – 15 and gain a deeper understanding of the underlying principles of algebra.

Step-by-Step Guide to Find the Solution Set for x^2 = 12x – 15

To solve the quadratic equation x^2 = 12x – 15 using the completing the square method, follow these steps:

1. Rearrange the equation to bring all terms to one side: x^2 – 12x + 15 = 0.
2. To complete the square, take half of the coefficient of x (in this case, 12) and square it to get 36. Add and subtract this value inside the parentheses: x^2 – 12x + 36 – 36 + 15 = 0.
3. Factor the perfect square trinomial: (x – 6)^2 – 36 + 15 = 0.
4. Simplify the equation: (x – 6)^2 – 21 = 0.
5. Add 21 to both sides: (x – 6)^2 = 21.
6. Solve for x by taking the square root of both sides and consider both positive and negative roots: x – 6 = ±√21.
7. Finally, find the solution set by adding 6 to both sides to obtain x = 6 ± √21.

By following these steps carefully, you can effectively solve the quadratic equation x^2 = 12x – 15 using the completing the square method. This approach not only provides you with the solution set but also enhances your problem-solving skills and algebraic reasoning. Practice completing the square on various quadratic equations to strengthen your understanding of this valuable technique and its applications in mathematics.