Factoring $2x^2+3x-54$: Find The Two Factors!

by Andrew McMorgan 48 views

Hey Plastik Magazine readers! Let's dive into some math and figure out how to factor the quadratic expression $2x^2+3x-54$. Factoring quadratics might seem tricky at first, but with a little practice, you'll get the hang of it. We're going to break down the steps and find the two correct factors from the options provided. So, grab your pencils and let's get started!

Understanding Factoring

Before we jump into this specific problem, let's quickly recap what factoring means. When we factor an expression, we're essentially trying to rewrite it as a product of two or more smaller expressions. Think of it like reverse multiplication. For example, if we multiply $(x + 2)$ and $(x + 3)$, we get $x^2 + 5x + 6$. So, factoring $x^2 + 5x + 6$ means finding $(x + 2)$ and $(x + 3)$. In this case, we want to find the two binomials that multiply together to give us $2x^2+3x-54$. This is a crucial concept, so make sure you have a solid grasp on it before moving forward. Understanding the basics will make the entire process smoother and more intuitive.

The Importance of Factoring in Mathematics

Factoring isn't just some abstract math concept; it's a fundamental skill with applications across various areas of mathematics. From solving quadratic equations to simplifying complex algebraic expressions, the ability to factor effectively is a powerful tool. For instance, when solving quadratic equations, factoring allows us to rewrite the equation in a form where we can easily find the roots or solutions. It's also essential in calculus, where factoring can simplify expressions and make differentiation and integration easier. Moreover, in fields like engineering and physics, factoring is used to analyze systems and solve problems involving polynomials. So, mastering factoring isn't just about acing your math exams; it's about building a strong foundation for future studies and career paths. By understanding and practicing factoring, you're equipping yourself with a skill that will prove invaluable in a wide range of contexts.

Steps to Factor the Quadratic Expression

Okay, let's tackle the expression $2x^2+3x-54$. Since the coefficient of the $x^2$ term is not 1, we'll use a slightly modified approach. Here's a step-by-step guide:

  1. Multiply the leading coefficient and the constant term: In our case, we multiply 2 (the coefficient of $x^2$) and -54 (the constant term) to get -108.
  2. Find two numbers that multiply to -108 and add up to 3 (the coefficient of the x term). This is the crucial step! We need to think of factor pairs of -108 that have a difference of 3. After some thought, we'll find that 12 and -9 fit the bill because 12 * -9 = -108 and 12 + (-9) = 3.
  3. Rewrite the middle term using these two numbers: We rewrite $3x$ as $12x - 9x$. So, our expression becomes $2x^2 + 12x - 9x - 54$.
  4. Factor by grouping: Now, we group the terms in pairs and factor out the greatest common factor (GCF) from each pair.
    • From the first two terms, $2x^2 + 12x$, we can factor out $2x$, which gives us $2x(x + 6)$.
    • From the last two terms, $-9x - 54$, we can factor out -9, which gives us $-9(x + 6)$.
  5. Factor out the common binomial: Notice that both terms now have a common factor of $(x + 6)$. We factor this out, and we're left with $(2x - 9)(x + 6)$.

And there you have it! We've factored the expression $2x^2+3x-54$ into $(2x - 9)(x + 6)$.

Breaking Down Each Step with Examples

Let's dive deeper into each step of the factoring process with some clear examples to make sure you've got a solid grasp on it. This will help clarify any confusion and give you the confidence to tackle similar problems on your own.

1. Multiply the Leading Coefficient and the Constant Term

The first step in factoring a quadratic expression like $ax^2 + bx + c$ is to multiply the leading coefficient (a) by the constant term (c). This product gives us a target number that we'll use to find the right combination of factors. For example, if we have the expression $2x^2 + 5x - 3$, we multiply 2 (the coefficient of $x^2$) by -3 (the constant term) to get -6. This -6 is the number we'll focus on in the next step.

  • Example 1: In the expression $3x^2 - 7x + 2$, we multiply 3 by 2 to get 6.
  • Example 2: For $4x^2 + 8x - 5$, we multiply 4 by -5 to get -20.

2. Find Two Numbers That Multiply to the Result and Add Up to the Middle Coefficient

Next, we need to find two numbers that not only multiply to the result we got in the previous step but also add up to the coefficient of the middle term (b). This step requires a bit of mental math and sometimes a bit of trial and error. Let's say our target number is -6 (from the example $2x^2 + 5x - 3$), and we want the two numbers to add up to 5 (the coefficient of the x term). The numbers 6 and -1 work perfectly because 6 * -1 = -6 and 6 + (-1) = 5.

  • Example 1: If our target is 6 (from $3x^2 - 7x + 2$) and we need the numbers to add up to -7, the numbers -6 and -1 fit the bill (-6 * -1 = 6, -6 + (-1) = -7).
  • Example 2: For a target of -20 (from $4x^2 + 8x - 5$) and a sum of 8, the numbers 10 and -2 work (10 * -2 = -20, 10 + (-2) = 8).

3. Rewrite the Middle Term Using These Two Numbers

Once we've found the magic numbers, we rewrite the middle term of our quadratic expression using these numbers. This process breaks the middle term into two parts, making it easier to factor by grouping. Using our example $2x^2 + 5x - 3$, we found the numbers 6 and -1. So, we rewrite 5x as 6x - x. Our expression now becomes $2x^2 + 6x - x - 3$.

  • Example 1: In $3x^2 - 7x + 2$, we found -6 and -1. Rewriting -7x gives us $3x^2 - 6x - x + 2$.
  • Example 2: With $4x^2 + 8x - 5$, using 10 and -2, we get $4x^2 + 10x - 2x - 5$.

4. Factor by Grouping

This step involves grouping the first two terms and the last two terms of our rewritten expression and factoring out the greatest common factor (GCF) from each group. It's like splitting the problem into two smaller factoring problems. Let's continue with $2x^2 + 6x - x - 3$. From the first group, $2x^2 + 6x$, we can factor out $2x$, leaving us with $2x(x + 3)$. From the second group, $-x - 3$, we can factor out -1, giving us $-1(x + 3)$.

  • Example 1: Factoring $3x^2 - 6x - x + 2$, we get $3x(x - 2) - 1(x - 2)$.
  • Example 2: For $4x^2 + 10x - 2x - 5$, we obtain $2x(2x + 5) - 1(2x + 5)$.

5. Factor Out the Common Binomial

The final step is to notice that both terms now have a common binomial factor. We factor this out, and we're left with our factored expression. In our example, we have $2x(x + 3) - 1(x + 3)$. Both terms have $(x + 3)$ as a common factor, so we factor it out, giving us $(2x - 1)(x + 3)$. And that's our factored expression!

  • Example 1: From $3x(x - 2) - 1(x - 2)$, we factor out $(x - 2)$ to get $(3x - 1)(x - 2)$.
  • Example 2: Factoring $(2x(2x + 5) - 1(2x + 5)$, we get $(2x - 1)(2x + 5)$.

By breaking down each step and illustrating with examples, we've made the factoring process much clearer and more manageable. Practice these steps, and you'll become a factoring pro in no time!

Identifying the Correct Options

Now that we've factored $2x^2+3x-54$ as $(2x - 9)(x + 6)$, we can easily identify the correct options from the list:

  • A. $2x - 9$: This is one of our factors!
  • B. $2x - 6$: This is not a factor.
  • C. $2x + 6$: This is not a factor.
  • D. $x - 6$: This is not a factor.
  • E. $x + 6$: This is the other factor we found!

So, the correct answers are A. $(2x - 9)$ and E. $(x + 6)$. Great job, guys! We nailed it!

Double-Checking Your Answer

Double-checking your answer is a crucial step in any math problem, and factoring is no exception. It's super easy to make a small mistake, especially when dealing with multiple steps and signs. So, how do we ensure our factored expression is correct? The simplest way is to multiply the factors back together and see if we get the original expression. If the multiplication checks out, we can be confident in our answer. Let's walk through how to do this.

Multiplying the Factors

To double-check our factored expression, we'll use the distributive property (often remembered by the acronym FOIL - First, Outer, Inner, Last). This method ensures we multiply each term in the first binomial by each term in the second binomial.

Let’s say we factored a quadratic expression and found the factors $(ax + b)$ and $(cx + d)$. To check our work, we multiply these two binomials:

(ax+b)(cx+d)=ax(cx)+ax(d)+b(cx)+b(d)(ax + b)(cx + d) = ax(cx) + ax(d) + b(cx) + b(d)

This gives us:

acx2+adx+bcx+bdacx^2 + adx + bcx + bd

Now, we combine like terms (the terms with the same power of x), which usually involves adding the $adx$ and $bcx$ terms:

acx2+(ad+bc)x+bdacx^2 + (ad + bc)x + bd

If this resulting expression matches our original quadratic expression, we’ve factored it correctly!

Example: Checking $(2x - 9)(x + 6)$

Let's use our factored expression from earlier, $(2x - 9)(x + 6)$, to demonstrate this process. We want to double-check that when we multiply these factors, we get back our original expression, $2x^2 + 3x - 54$.

  1. Multiply the First terms: $(2x)(x) = 2x^2$
  2. Multiply the Outer terms: $(2x)(6) = 12x$
  3. Multiply the Inner terms: $(-9)(x) = -9x$
  4. Multiply the Last terms: $(-9)(6) = -54$

Now, we add these products together:

2x2+12x−9x−542x^2 + 12x - 9x - 54

Combine the like terms (the x terms):

2x2+(12x−9x)−542x^2 + (12x - 9x) - 54

2x2+3x−542x^2 + 3x - 54

Lo and behold, we got our original expression back! This confirms that our factored form, $(2x - 9)(x + 6)$, is indeed correct. This simple check gives us peace of mind and ensures we haven't made any errors.

Why Double-Checking is Essential

Double-checking is more than just a good habit; it's an essential part of problem-solving. It catches any small errors that might have slipped through during the factoring process. For example, a sign error or a mistake in multiplication can lead to an incorrect factored form. By multiplying the factors back together, we ensure that every term is accounted for and that the signs are correct. This not only verifies our answer but also reinforces our understanding of the factoring process.

In addition to catching errors, double-checking builds confidence. Knowing that you've verified your answer makes you more comfortable with your solution. This is especially important in exams or when applying factoring in more complex problems. The more you practice double-checking, the more natural it becomes, and the more confident you'll be in your math abilities.

By consistently double-checking your work, you're not just getting the right answer; you're also solidifying your understanding and improving your problem-solving skills. So, next time you factor a quadratic expression, remember to take that extra minute to multiply the factors back together – it’s totally worth it!

Conclusion

Factoring quadratic expressions can be a breeze once you understand the steps and practice them a bit. In this case, we successfully factored $2x^2+3x-54$ and identified the correct factors as $(2x - 9)$ and $(x + 6)$. Keep practicing, and you'll become a factoring master in no time! Remember, math is like a muscle – the more you use it, the stronger it gets. Keep challenging yourselves, and you'll be amazed at what you can achieve. Until next time, keep those pencils sharp and those brains engaged!