Which Expression Is Equivalent To 6x2-19x-55

Which Expression is Equivalent to 6x² - 19x - 55? A Comprehensive Guide to Factoring Quadratic Expressions

Finding equivalent expressions is a fundamental skill in algebra, particularly when dealing with quadratic equations. This comprehensive guide will explore various methods to determine which expression is equivalent to 6x² - 19x - 55, providing a deep understanding of factoring quadratic expressions and the underlying mathematical principles. We’ll cover different approaches, from traditional factoring techniques to the quadratic formula, and highlight the importance of checking your work to ensure accuracy.

Understanding Quadratic Expressions

Before diving into the solution, let's briefly review what a quadratic expression is. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable (in this case, x) is 2. It generally takes the form:

ax² + bx + c

where 'a', 'b', and 'c' are constants. In our problem, a = 6, b = -19, and c = -55. Our goal is to find an equivalent expression that is factored, meaning it's expressed as a product of simpler expressions.

Method 1: Factoring by Grouping

This method involves breaking down the middle term (-19x) into two terms whose sum is -19x and whose product is equal to the product of the first (6x²) and last (-55) terms.

Find the product of 'a' and 'c': 6 * -55 = -330
Find two numbers that add up to 'b' (-19) and multiply to -330: After some trial and error (or using a systematic approach), we find that -33 and 10 satisfy these conditions (-33 + 10 = -19 and -33 * 10 = -330).
Rewrite the middle term: Rewrite the expression as: 6x² - 33x + 10x - 55
Factor by grouping: Group the terms in pairs and factor out the greatest common factor (GCF) from each pair:

3x(2x - 11) + 5(2x - 11)
Factor out the common binomial: Notice that (2x - 11) is common to both terms. Factor it out:

(2x - 11)(3x + 5)

Therefore, the factored equivalent expression of 6x² - 19x - 55 is (2x - 11)(3x + 5).

Method 2: Using the Quadratic Formula

The quadratic formula provides a direct way to find the roots (solutions) of a quadratic equation, which can then be used to factor the expression. The quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

Identify a, b, and c: As before, a = 6, b = -19, and c = -55.
Substitute into the quadratic formula:

x = [19 ± √((-19)² - 4 * 6 * -55)] / (2 * 6)
Simplify:

x = [19 ± √(361 + 1320)] / 12 x = [19 ± √1681] / 12 x = [19 ± 41] / 12
Find the roots:

x₁ = (19 + 41) / 12 = 60 / 12 = 5 x₂ = (19 - 41) / 12 = -22 / 12 = -11/6
Construct the factors: The roots represent the values of x that make the expression equal to zero. We can use these roots to construct the factors:

(x - 5) and (x + 11/6)
Adjust for the leading coefficient: Since we have a leading coefficient of 6, we need to incorporate that into the factors. Multiplying the second factor by 6 to eliminate fractions we get:

6(x + 11/6) = 6x + 11.
The factored expression: (x - 5)(6x+11) is equal to 6x² - 19x -55

Notice that this is equivalent to (2x - 11)(3x + 5) obtained by the grouping method if you expand both expressions. This is just a different way of factoring the same expression, they are both completely equivalent.

Method 3: Trial and Error

This method involves systematically testing different combinations of factors of 'a' and 'c' until you find a combination that produces the correct middle term. While less systematic than the other methods, it can be quicker for simpler quadratic expressions. It involves trying various combinations of factors of 6 (1, 6; 2, 3) and factors of -55 (1, -55; 5, -11; -5, 11; -1, 55) and arranging them in the binomial pairs until the desired middle term is obtained.

Verifying Your Solution

Regardless of the method used, it's crucial to verify your answer by expanding the factored expression. Expanding (2x - 11)(3x + 5) gives:

6x² + 10x - 33x - 55 = 6x² - 19x - 55

This confirms that our factored expression is indeed equivalent to the original quadratic expression.

Common Mistakes to Avoid

Incorrect signs: Pay close attention to the signs when factoring. A small error in the sign can lead to an incorrect factored expression.
Incorrect GCF: Always ensure you've factored out the greatest common factor from each group when using the grouping method.
Not checking your work: Always expand your factored expression to verify that it matches the original expression.
Forgetting the leading coefficient: When using the quadratic formula, remember to adjust the factors to reflect the leading coefficient of the original quadratic.

Applications of Factoring Quadratic Expressions

Factoring quadratic expressions is a crucial skill with numerous applications in various areas of mathematics and beyond:

Solving quadratic equations: Factoring allows you to find the roots (solutions) of quadratic equations, which are essential in many fields, including physics, engineering, and finance.
Graphing quadratic functions: The factored form of a quadratic expression helps determine the x-intercepts (where the graph crosses the x-axis) of the corresponding parabola.
Simplifying algebraic expressions: Factoring can simplify complex algebraic expressions, making them easier to work with.
Calculus: Factoring is used extensively in calculus for differentiation and integration.

Conclusion

Determining which expression is equivalent to 6x² - 19x - 55 involves understanding the fundamentals of factoring quadratic expressions. Whether you employ factoring by grouping, the quadratic formula, or trial and error, meticulous attention to detail and verification of the result are paramount. Mastering these techniques is not only beneficial for solving algebraic problems but also lays a strong foundation for more advanced mathematical concepts. Remember to practice regularly to build your proficiency and confidence in handling quadratic expressions and other algebraic manipulations. The more you practice, the quicker and more intuitive this process will become.