How to Factorize Quadratic Expressions: A Step-by-Step Guide

Factoring quadratic expressions is a fundamental algebraic skill that is widely used in mathematics, engineering, and other scientific fields. In this article, we will walk through the process of factoring quadratic expressions, specifically focusing on the factorization of perfect square trinomials.

Understanding the Basics

Quadratic expressions take the form of (ax^2 bx c). To factorize a quadratic expression, we often look for a way to rewrite it as a product of binomials. One common type of quadratic expression is the perfect square trinomial.

Identifying a Perfect Square Trinomial

A perfect square trinomial is a quadratic expression of the form (a^2 - 2ab b^2) or (a^2 2ab b^2). These expressions can be factored as ((a - b)^2) or ((a b)^2) respectively.

Example: Factorizing (81b^2 - 90ab 25a^2)

Let's consider the quadratic expression 81b^2 - 90ab 25a^2.

Step 1: Recognize the Structure

First, we need to identify if the expression is a perfect square trinomial. We will look at the first and last terms:

( (25a^2 (5a)^2)

Next, we check if the middle term is two times the product of the square roots of the first and last terms:

(-90ab 2 times (9b) times (-5a))

This confirms that the expression is a perfect square trinomial.

Step 2: Apply the Formula

Using the formula for a perfect square trinomial, we can rewrite the expression as:

[(9b - 5a)^2]

Thus, the factored form of the expression (81b^2 - 90ab 25a^2) is (9b - 5a)^2.

Additional Example: Factorizing (81b^2 - 90ab 25a^2 0)

Now, let's consider the quadratic equation 81b^2 - 90ab 25a^2 0.

Step 1: Rewrite the Equation

Let's rewrite the equation in a simplified form:

[(9b - 5a)^2 0]

Step 2: Solve for the Variables

Since ((9b - 5a)^2 0), we can take the square root of both sides:

[9b - 5a 0]

Therefore, the solution to the equation is:

[9b 5a] [b frac{5a}{9}]

This means that the values of (b) and (a) must satisfy the equation (b frac{5a}{9}).

Conclusion

In conclusion, factoring quadratic expressions, especially perfect square trinomials, is a valuable skill in algebra. By recognizing the structure of the expression and applying the appropriate formula, we can efficiently factorize the expression and, in some cases, solve the corresponding equation.

Related Keywords

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Further Reading

For more in-depth information and tutorials on factoring quadratic expressions, visit various math education websites and resources.