Factoring 4x² - 13x + 10 A Step By Step Guide

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Factoring quadratic expressions is a fundamental skill in algebra, and it's essential for solving equations, simplifying expressions, and understanding the behavior of quadratic functions. In this article, we will delve into the process of factoring the quadratic expression 4x² - 13x + 10. We'll break down the steps, explain the reasoning behind them, and provide a clear and concise solution. Understanding how to factor this type of expression will equip you with a valuable tool for tackling more complex algebraic problems.

Understanding Quadratic Expressions

Before we dive into the specifics of factoring 4x² - 13x + 10, it's crucial to grasp the general form of a quadratic expression. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable is two. The standard form of a quadratic expression is:

ax² + bx + c

Where:

  • a, b, and c are constants (numbers).
  • x is the variable.
  • a is the coefficient of the x² term.
  • b is the coefficient of the x term.
  • c is the constant term.

In our case, for the expression 4x² - 13x + 10:

  • a = 4
  • b = -13
  • c = 10

Factoring a quadratic expression involves rewriting it as a product of two linear expressions (expressions where the highest power of the variable is one). This is the reverse process of expanding two binomials (expressions with two terms) using the distributive property (also known as FOIL method).

The Factoring Process: A Step-by-Step Approach

There are several methods for factoring quadratic expressions, but we'll focus on the most common technique: the ac method or the factoring by grouping method. This method is particularly useful when the leading coefficient (a) is not equal to 1.

Let's break down the steps involved in factoring 4x² - 13x + 10 using the ac method:

Step 1: Multiply 'a' and 'c'

Our first step is to multiply the coefficient of the x² term (a) by the constant term (c). In our expression, a = 4 and c = 10, so:

4 * 10 = 40

This product, 40, will be the key to finding the right factors.

Step 2: Find Two Factors of 'ac' That Add Up to 'b'

Now, we need to find two numbers that multiply to give us the product we calculated in Step 1 (40) and add up to the coefficient of the x term (b), which is -13. This is often the most challenging part of the process, and it may require some trial and error.

Let's list the factors of 40 (both positive and negative) and see which pair sums to -13:

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

We can see that the pair -5 and -8 satisfies both conditions:

  • (-5) * (-8) = 40
  • (-5) + (-8) = -13

These two numbers, -5 and -8, are crucial for the next step.

Step 3: Rewrite the Middle Term ('bx') Using the Factors

Now we rewrite the middle term (-13x) of our quadratic expression using the two factors we found (-5 and -8). This means we replace -13x with -5x - 8x. Our expression now looks like this:

4x² - 5x - 8x + 10

Notice that we haven't changed the value of the expression; we've simply rewritten it in a way that allows us to factor by grouping.

Step 4: Factor by Grouping

This is where the “grouping” part of the method comes into play. We divide the expression into two groups:

(4x² - 5x) + (-8x + 10)

Now, we factor out the greatest common factor (GCF) from each group. In the first group (4x² - 5x), the GCF is x. Factoring out x, we get:

x(4x - 5)

In the second group (-8x + 10), the GCF is -2 (it's important to factor out a negative number here to make the expressions inside the parentheses match). Factoring out -2, we get:

-2(4x - 5)

Now our expression looks like this:

x(4x - 5) - 2(4x - 5)

Notice that both terms now have a common factor of (4x - 5). This is a key indication that we're on the right track.

Step 5: Factor Out the Common Binomial Factor

We now factor out the common binomial factor (4x - 5) from the entire expression:

(4x - 5)(x - 2)

And that's it! We have successfully factored the quadratic expression 4x² - 13x + 10.

The Solution and Verification

The factored form of the expression 4x² - 13x + 10 is:

(4x - 5)(x - 2)

To verify that our factoring is correct, we can expand the factored form using the distributive property (FOIL method) and see if we get back our original expression:

(4x - 5)(x - 2) = 4x(x) + 4x(-2) - 5(x) - 5(-2) = 4x² - 8x - 5x + 10 = 4x² - 13x + 10

Since we obtained our original expression, we can be confident that our factoring is correct.

Common Mistakes to Avoid

Factoring quadratic expressions can be tricky, and there are some common mistakes that students often make. Here are a few to watch out for:

  • Incorrectly identifying factors: The most common mistake is choosing the wrong factors in Step 2. Always double-check that the factors multiply to 'ac' and add up to 'b'.
  • Sign errors: Pay close attention to the signs of the factors and the terms in the expression. A simple sign error can lead to an incorrect factorization.
  • Forgetting to factor out the GCF: Before attempting to factor a quadratic expression, always look for a greatest common factor (GCF) that can be factored out from all the terms. This simplifies the factoring process.
  • Stopping too early: Make sure you have factored the expression completely. Double-check that the resulting factors cannot be factored further.

Practice Makes Perfect

The best way to master factoring quadratic expressions is through practice. Work through a variety of examples, and don't be discouraged if you make mistakes. Each mistake is a learning opportunity. Try factoring the following expressions on your own:

  • 2x² + 7x + 3
  • 3x² - 10x + 8
  • 5x² + 13x - 6

By consistently practicing, you'll develop a strong understanding of the factoring process and be able to tackle even the most challenging quadratic expressions with confidence.

Conclusion

Factoring quadratic expressions is a crucial skill in algebra and has wide-ranging applications in mathematics and other fields. By understanding the steps involved in the ac method (factoring by grouping) and practicing regularly, you can master this skill and enhance your problem-solving abilities. Remember to double-check your work, avoid common mistakes, and embrace the challenge of factoring – it's a rewarding journey that will strengthen your mathematical foundation.

In this article, we have successfully factored the expression 4x² - 13x + 10 into (4x - 5)(x - 2). This process involved multiplying the leading coefficient and the constant term, finding factors that add up to the middle term's coefficient, rewriting the middle term, and factoring by grouping. By following these steps carefully, you can factor a wide range of quadratic expressions and unlock new levels of algebraic understanding. Now go forth and conquer those quadratic equations!