Factoring Quadratic Expressions How To Solve 16y² + 12y - 10

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Understanding Quadratic Expressions

Before diving into the factorization, let's understand the basics. A quadratic expression is a polynomial of degree two, generally represented in the form ax² + bx + c, where a, b, and c are constants, and 'x' is the variable. In our case, the expression is 16y² + 12y - 10, where a = 16, b = 12, and c = -10. The variable here is 'y' instead of 'x', but the principles remain the same. Factoring a quadratic expression involves breaking it down into a product of two linear expressions.

The significance of factoring lies in its ability to simplify complex algebraic problems. When a quadratic expression is factored, it becomes easier to find the roots of the corresponding quadratic equation (by setting the expression equal to zero). Factoring also helps in simplifying rational expressions and solving various application problems in mathematics and science. Mastering factoring techniques is therefore crucial for any student studying algebra.

To effectively factor quadratic expressions, one needs to understand the relationship between the coefficients (a, b, and c) and the factors. The factors are typically in the form (px + q)(rx + s), where p, q, r, and s are constants. Expanding this form gives us prx² + (ps + qr)x + qs. By comparing this with the general form ax² + bx + c, we can see that pr = a, ps + qr = b, and qs = c. These relationships guide the factoring process, helping us find the correct combination of factors.

Step-by-Step Factorization of 16y² + 12y - 10

Now, let's factor the given expression, 16y² + 12y - 10, step by step:

Step 1: Look for a Common Factor

The first step in factoring any expression is to check for a common factor among all the terms. In our expression, 16y² + 12y - 10, we can see that all coefficients (16, 12, and -10) are divisible by 2. Factoring out the common factor 2, we get:

2(8y² + 6y - 5)

This simplifies the expression inside the parentheses and makes it easier to factor further. Always look for a common factor first, as it simplifies the subsequent steps.

Step 2: Factor the Quadratic Expression Inside the Parentheses

Now we need to factor the quadratic expression 8y² + 6y - 5. This is a trinomial of the form ay² + by + c, where a = 8, b = 6, and c = -5. To factor this, we look for two numbers that multiply to ac (8 * -5 = -40) and add up to b (6). This method is often called the "ac method".

Let's find the pairs of factors of -40:

  • -1 and 40
  • -2 and 20
  • -4 and 10
  • -5 and 8
  • -8 and 5
  • -10 and 4
  • -20 and 2
  • -40 and 1

Among these pairs, -4 and 10 add up to 6, which is our b value. So, we will use these numbers to split the middle term.

Step 3: Split the Middle Term

We rewrite the middle term (6y) using the numbers we found (-4 and 10):

8y² + 6y - 5 = 8y² - 4y + 10y - 5

By splitting the middle term, we've converted the trinomial into a four-term expression, which can be factored by grouping.

Step 4: Factor by Grouping

Now we group the first two terms and the last two terms:

(8y² - 4y) + (10y - 5)

Factor out the greatest common factor (GCF) from each group:

4y(2y - 1) + 5(2y - 1)

Notice that both groups now have a common factor of (2y - 1). We can factor this out:

(2y - 1)(4y + 5)

This is the factored form of the quadratic expression 8y² + 6y - 5.

Step 5: Combine the Factors

Remember that we initially factored out a 2 in Step 1. Now we need to include that factor in our final result. So, the factored form of the original expression 16y² + 12y - 10 is:

2(2y - 1)(4y + 5)

This is the complete factorization of the given quadratic expression.

Analyzing the Options and Finding the Correct Answer

Now that we have factored the expression, let's compare our result with the given options:

a) (4y - 5)(4y - 2) b) (4y + 2)(4y + 5) c) (4y - 2)(4y + 5) d) (4y - 5)(4y + 2)

Our factored form is 2(2y - 1)(4y + 5). To match this with the options, we need to distribute the 2 into one of the factors. If we distribute the 2 into the first factor (2y - 1), we get (4y - 2). Thus, our factored form can also be written as:

(4y - 2)(4y + 5)

Comparing this with the options, we find that option c) matches our result. Therefore, the correct answer is:

c) (4y - 2)(4y + 5)

It's crucial to note that the order of the factors does not matter since multiplication is commutative (a * b = b * a). So, (4y - 2)(4y + 5) is the same as (4y + 5)(4y - 2).

Common Mistakes to Avoid

Factoring quadratic expressions can be tricky, and it's easy to make mistakes. Here are some common mistakes to watch out for:

  • Forgetting to factor out the greatest common factor (GCF): Always look for a GCF first. If you don't, you might end up with a more complex expression to factor.
  • Incorrectly identifying the factors: Make sure the numbers you choose multiply to ac and add up to b. Double-check your calculations.
  • Sign errors: Pay close attention to the signs of the terms. A small sign error can lead to an incorrect factorization.
  • Not checking your answer: After factoring, you can always check your answer by multiplying the factors back together. If you get the original expression, your factorization is correct.
  • Distributing the common factor incorrectly: When you factor out a common factor, make sure you distribute it back correctly when comparing with the options.

By being aware of these common mistakes, you can improve your accuracy and avoid pitfalls in factoring quadratic expressions.

Advanced Factoring Techniques

While the "ac method" is widely used, there are other techniques that can be helpful in factoring quadratic expressions. Here are a couple of them:

Factoring by Perfect Square Trinomials

A perfect square trinomial is a quadratic expression that can be written in the form (ax + b)² or (ax - b)². These trinomials have a specific pattern:

a²x² + 2abx + b² = (ax + b)² a²x² - 2abx + b² = (ax - b)²

If you recognize this pattern in a quadratic expression, you can factor it directly without going through the ac method.

Factoring the Difference of Squares

The difference of squares is a binomial expression of the form a² - b², which can be factored as:

a² - b² = (a + b)(a - b)

This pattern is particularly useful when dealing with expressions that have two terms, both of which are perfect squares and are separated by a minus sign.

Factoring by Substitution

In some cases, quadratic expressions can be disguised within more complex expressions. In such cases, substitution can simplify the problem. For example, if you have an expression like (x² + 1)² + 5(x² + 1) + 6, you can substitute y = x² + 1 to get y² + 5y + 6, which is easier to factor.

By mastering these advanced techniques, you'll be able to tackle a wider range of factoring problems with greater efficiency.

Practice Problems

To solidify your understanding of factoring quadratic expressions, it's essential to practice. Here are a few practice problems:

  1. Factor 6x² + 11x - 10
  2. Factor 9y² - 25
  3. Factor 4x² + 20x + 25
  4. Factor 2x² - 18
  5. Factor 3x² + 10x - 8

Try solving these problems using the techniques we've discussed. Remember to always look for a common factor first and double-check your answers.

Conclusion

Factoring quadratic expressions is a crucial skill in algebra. By understanding the underlying principles and following a systematic approach, you can master this skill and solve a wide range of problems. In this guide, we've walked through the step-by-step factorization of 16y² + 12y - 10, analyzed the options, and identified the correct answer. We've also discussed common mistakes to avoid, advanced factoring techniques, and provided practice problems to help you solidify your understanding. With consistent practice, you'll become proficient in factoring quadratic expressions and be well-prepared for more advanced topics in mathematics.