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Completing the Square: A Step-by-Step Guide
In the realm of algebra, quadratic equations play a crucial role. These equations, characterized by their highest power of two, often require specific techniques for finding their solutions. One such technique, known as completing the square, provides a systematic approach to solving quadratic equations and offers a deeper understanding of their structure.
Understanding the Concept
Completing the square is a method that transforms a quadratic equation into a perfect square trinomial, which can then be easily solved by taking the square root of both sides. The key principle behind this technique lies in manipulating the equation to create a trinomial that can be factored into the square of a binomial.
Step-by-Step Guide
Let’s illustrate the process of completing the square with a step-by-step example:
1. Rearrange the Equation
Begin by rearranging the quadratic equation so that the constant term is isolated on one side of the equation. For instance, consider the equation:
x² + 6x – 7 = 0
Move the constant term to the right side:
x² + 6x = 7
2. Complete the Square
To complete the square, we focus on the left side of the equation. Take half of the coefficient of the x term (which is 6 in this case), square it (6/2 = 3, 3² = 9), and add it to both sides of the equation:
x² + 6x + 9 = 7 + 9
The left side now represents a perfect square trinomial:
(x + 3)² = 16
3. Apply the Square Root Property
Take the square root of both sides of the equation:
√(x + 3)² = ±√16
Simplify:
x + 3 = ±4
4. Solve for x
Isolate x by subtracting 3 from both sides:
x = -3 ± 4
Therefore, the solutions to the quadratic equation are:
x = 1 or x = -7
Applications of Completing the Square
Completing the square is a versatile technique with applications in various mathematical contexts, including:
- Solving Quadratic Equations: It provides a systematic method for finding the roots or solutions of quadratic equations.
- Deriving the Quadratic Formula: The quadratic formula, a general solution for quadratic equations, can be derived using the method of completing the square.
- Graphing Parabolas: Completing the square helps determine the vertex and axis of symmetry of a parabola, facilitating its accurate graphing.
- Calculus: Completing the square is used in calculus for simplifying integrals and finding the maxima and minima of functions.
Conclusion
Completing the square is a powerful technique for solving quadratic equations and gaining a deeper understanding of their structure. By systematically manipulating the equation, we can transform it into a perfect square trinomial, allowing for efficient solution. This method finds applications in various areas of mathematics and is a valuable tool for any student of algebra.