Factorizing Quadratics: A Step-by-Step Guide to 17x2 - 68

Factorization, a fundamental concept in algebra, involves breaking down an expression into simpler terms that, when multiplied together, give back the original expression. In this article, we will explore the process of factorizing a quadratic equation, using the example 17x2 - 68.

Introduction to Factorization of Quadratic Expressions

Quadratic equations are algebraic expressions of the form ax2 bx c 0, where a, b, and c are constants and a ≠ 0. Understanding how to factorize such equations is crucial for solving them and simplifying complex algebraic manipulations. Here, we will demonstrate the factorization of the given quadratic expression and highlight the key steps involved.

Step-by-Step Factorization of 17x2 - 68

To factorize the quadratic expression 17x2 - 68, we start by examining the structure of the expression. The expression can be represented as:

17x2 - 68 0

Step 1: Divide the Entire Expression by the Coefficient of x2

First, we divide the entire expression by the coefficient of x2, which is 17:

(17x2 - 68) / 17 x2 - 4

This simplification leads us to the expression x2 - 4.

Step 2: Recognize the Pattern - Difference of Squares

The expression x2 - 4 is recognizable as a difference of squares. The difference of squares formula states that a2 - b2 (a b)(a - b). In our case, x2 - 4 can be rewritten as:

x2 - 4 (x 2)(x - 2)

Step 3: Verify the Factorization

To ensure the factorization is correct, we can expand the factored form and verify it matches the original expression:

(x 2)(x - 2) x2 - 2x 2x - 4 x2 - 4

Thus, the factorization is verified.

Step 4: Solve the Quadratic Equation

Given the equation x2 - 4 0, we can solve for x:

x2 - 4 0

x2 4

x ±2

Therefore, the solutions to the equation are x 2 and x -2.

Key Concepts in Quadratic Factorization

Understanding the factorization of quadratic expressions is essential for solving a wide range of mathematical problems, including: Solving Quadratic Equations: Factoring provides a quicker method to find the roots of a quadratic equation. Graphing Quadratic Functions: Factorized form can help in identifying the x-intercepts of the quadratic function. Algebraic Manipulation: Factorization is used to simplify or solve more complex algebraic expressions.

Additional Resources

If you're looking to deepen your understanding of quadratic factorization and other algebraic concepts, consider exploring these resources: Lamar University's Algebra Tutorial Khan Academy’s Quadratic Equations Section Math is Fun: Quadratic Equations

Conclusion

Mastering the technique of factorizing quadratic expressions, such as 17x2 - 68, opens up a pathway to solving more complex algebraic problems. Through the application of initial simplification, recognizing the difference of squares, and solving step-by-step, you can unlock the solution more efficiently.