Choosing the Best Method to Factor a Quadratic Expression

In this article, we will explore the process of factoring the quadratic expression 5x2 - 8x 3. We will discuss the best method to use for this expression and explain why it is the most efficient approach. The options we will consider are decomposition, trial and error, and the quadratic formula. We will also delve into the step-by-step process of using decomposition and provide additional insights into the other methods.

Understanding Quadratic Expressions

Quadratic expressions are polynomial expressions of degree 2, represented as ax2 bx c. In the given expression, 5x2 - 8x 3, the coefficients are as follows: A5, B-8, C3.

Best Method for Factoring: Decomposition

The best method to factor the given expression is decomposition. Here's why:

Efficiency: Decomposition is efficient when the leading coefficient (a) is not 1, as in this case. It is a structured and less time-consuming approach compared to trial and error. Systematic: Decomposition provides a clear and systematic way to factor the expression, making it easier to follow and apply. Reliability: For quadratics where the leading coefficient is not 1, decomposition is often more effective than trial and error.

Step-by-Step Process of Decomposition

Let's walk through the steps to factor the expression using decomposition:

Multiply the Leading Coefficient and the Constant Term: Multiply the leading coefficient (5) by the constant term (3).

5 * 3  15

Find Two Numbers that Multiply to 15 and Add to -8: Find two numbers that multiply to 15 and add to -8. These numbers are -3 and -5.

-3 * -5  15-3   -5  -8

Rewrite the Middle Term Using These Numbers:

5x2 - 3x - 5x   3

Group the Terms:

(5x2 - 3x)   (-5x   3)

Factor by Grouping:

x(5x - 3) - 1(5x - 3)

Factor Out the Common Binomial:

(5x - 3)(x - 1)

Therefore, the factored form of the quadratic expression 5x2 - 8x 3 is (5x - 3)(x - 1).

Alternative Methods: Trial and Error and Quadratic Formula

While decomposition is the most efficient method for the given quadratic expression, it is interesting to consider the other methods:

Trial and Error

Trial and error is an approach where one guesses the factors of the quadratic expression. However, this method can be less systematic and time-consuming, especially when dealing with larger or more complex coefficients. For the given expression, the trial and error method would involve guessing the factors, but it is often less efficient than decomposition.

Quadratic Formula

The quadratic formula is a reliable method for finding the roots of any quadratic equation. It involves solving the equation x [-b ± sqrt(b2 - 4ac)] / (2a). However, while it is reliable, it does not directly yield the factored form. Using the quadratic formula:

Calculate the Discriminant: [b^2 - 4ac (-8)^2 - 4(5)(3) 64 - 60 4] Solve for x:

x  [-(-8) ± sqrt(4)] / (2 * 5)   [8 ± 2] / 10   1 or 0.6

Write in Factored Form: The roots are x 1 and x 0.6, so the factored form is (x - 1)(5x - 3) or (x - 0.6)(5x - 3).

While the quadratic formula is reliable, it involves additional steps to write the expression in factored form.

Conclusion

For factoring the quadratic expression 5x2 - 8x 3, decomposition is the best method due to its efficiency and systematic nature. It is particularly useful when the leading coefficient is not 1. While trial and error and the quadratic formula are reliable, they may involve additional steps or less systematic approaches.