Perfect Squares and Completing the Square: Techniques and Examples

Introduction to Perfect Squares

A perfect square is a quadratic expression of the form (ax^2 bx c) that can be expressed as the square of a binomial. Examples include expressions like (2x^2 5x 6.25), which can be factored as ((2x 2.5)^2).

Completing the Square: A Method to Form a Perfect Square

To complete the square of a quadratic expression (ax^2 bx), follow these steps:

Factor out the coefficient (a) from the first two terms: Complete the square for the expression inside the parentheses: Add and subtract the square of half the coefficient of (x): Simplify the expression:

Let's apply this method to the quadratic expression 2x^2 5x to make it a perfect square:

Completing the Square: Example 1

Given expression: (2x^2 5x)

1. Factor out the coefficient 2:

(2(x^2 frac{5}{2}x))

2. Complete the square for (x^2 frac{5}{2}x):

(x^2 frac{5}{2}x (x frac{5}{4})^2 - (frac{5}{4})^2)

3. Substitute back into the original expression:

(2(x frac{5}{4})^2 - 2(frac{5}{4})^2)

4. Simplify:

(2(x frac{5}{4})^2 - frac{25}{8})

Thus, the expression (2x^2 5x) can be written as (2(x frac{5}{4})^2 - frac{25}{8}).

Adding a Constant to Make it a Perfect Square

If you want to add a constant (p) to the expression to make it a perfect square, follow these steps:

Formulate the equation to include a constant: Solve for the constant (p):

Given the expression (2x^2 2x), let's find (p) to make it a perfect square:

Adding a Constant to Make it a Perfect Square: Example 2

Given expression: (2x^2 2x p)

1. Let the perfect square be:

(2x^2 2x p 2(x 0.5)^2)

2. Simplify and solve for (p):

[2(x 0.5)^2 2(x^2 x 0.25) 2x^2 2x 0.5]

[p 0.5]

To make (2x^2 2x) a perfect square, add (0.5).

Conclusion

By understanding and applying these techniques, you can convert a quadratic expression into a perfect square. This process is useful in solving equations, graphing, and various applications in mathematics.