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The concept of perfect squares plays a crucial role in simplifying and solving quadratic equations. By understanding and applying this concept, we can efficiently factorize quadratic expressions and determine the roots of quadratic equations. This approach not only streamlines the solution process but also provides a deeper understanding of the underlying mathematical principles. Let's delve into the application of perfect squares in quadratic equations.

Understanding Perfect Squares

A perfect square is a number that can be obtained by squaring an integer. For instance, 9 is a perfect square because it is the result of squaring 3 (3² = 9). In the context of quadratic equations, a perfect square trinomial is a trinomial that can be factored into the square of a binomial. The general form of a perfect square trinomial is (ax + b)², where 'a' and 'b' are constants. Expanding this expression, we get a²x² + 2abx + b².

Applying Perfect Squares to Quadratic Equations

To apply the concept of perfect squares to quadratic equations, we need to manipulate the equation to create a perfect square trinomial. This can be achieved by completing the square. Completing the square involves adding a constant term to both sides of the equation to make the left-hand side a perfect square trinomial.

Let's consider a quadratic equation in the standard form: ax² + bx + c = 0. To complete the square, we follow these steps:

1. Divide both sides by 'a': This ensures that the coefficient of the x² term is 1.

2. Move the constant term to the right-hand side: This isolates the x² and x terms on the left-hand side.

3. Take half of the coefficient of the x term, square it, and add it to both sides: This step creates the perfect square trinomial on the left-hand side.

4. Factor the perfect square trinomial: The left-hand side can now be factored as (x + h)², where 'h' is half of the coefficient of the x term.

5. Solve for 'x': Take the square root of both sides and solve for 'x'.

Example: Solving a Quadratic Equation by Completing the Square

Let's solve the quadratic equation x² + 6x + 5 = 0 using the method of completing the square.

1. Divide both sides by 1: x² + 6x + 5 = 0

2. Move the constant term to the right-hand side: x² + 6x = -5

3. Take half of the coefficient of the x term (6/2 = 3), square it (3² = 9), and add it to both sides: x² + 6x + 9 = -5 + 9

4. Factor the perfect square trinomial: (x + 3)² = 4

5. Solve for 'x': √(x + 3)² = ±√4

x + 3 = ±2

x = -3 ± 2

x = -1 or x = -5

Therefore, the solutions to the quadratic equation x² + 6x + 5 = 0 are x = -1 and x = -5.

Conclusion

The application of perfect squares in quadratic equations provides a powerful tool for solving and understanding these equations. By completing the square, we can transform a quadratic equation into a form that allows us to easily determine its roots. This method not only simplifies the solution process but also enhances our comprehension of the underlying mathematical principles. The ability to recognize and manipulate perfect squares is an essential skill in algebra and has wide-ranging applications in various fields of mathematics and beyond.