Factor X 2 4x 21

Unraveling the Mystery: Factoring x² + 4x - 21

Factoring quadratic expressions like x² + 4x - 21 is a fundamental skill in algebra. Understanding this process opens doors to solving quadratic equations, graphing parabolas, and tackling more advanced mathematical concepts. This comprehensive guide will walk you through factoring this specific expression, explaining the underlying principles and providing strategies for tackling similar problems. We'll explore different methods, delve into the underlying mathematics, and address common questions, ensuring you gain a firm grasp of this important algebraic technique.

Understanding Quadratic Expressions

Before diving into the specifics of factoring x² + 4x - 21, let's establish a foundational understanding of quadratic expressions. A quadratic expression is a polynomial of degree two, meaning the highest power of the variable (usually x) is 2. The general form of a quadratic expression is ax² + bx + c, where a, b, and c are constants (numbers). In our case, a = 1, b = 4, and c = -21.

The goal of factoring a quadratic expression is to rewrite it as a product of two simpler expressions, usually binomials. This process is crucial because it allows us to find the roots (or zeros) of the corresponding quadratic equation (ax² + bx + c = 0), which represent the x-intercepts of the parabola when graphed.

Method 1: Factoring by Inspection (Trial and Error)

This method relies on understanding the relationship between the factors of a and c and their relationship to b. Since our expression is x² + 4x - 21, we're looking for two binomials that multiply to give us this quadratic. We know the binomials will be in the form (x + p)(x + q), where p and q are numbers.

The key is that:

• p * q = c (-21): The product of p and q must equal the constant term, -21.
• p + q = b (4): The sum of p and q must equal the coefficient of the x term, 4.

Let's find the pairs of factors of -21:

• (-1, 21) Their sum is 20.
• (1, -21) Their sum is -20.
• (-3, 7) Their sum is 4.
• (3, -7) Their sum is -4.

We see that -3 and 7 satisfy both conditions: (-3) * 7 = -21 and (-3) + 7 = 4. Therefore, the factored form of x² + 4x - 21 is (x - 3)(x + 7).

Method 2: AC Method

The AC method, also known as the grouping method, provides a more systematic approach, especially useful when dealing with more complex quadratic expressions where trial and error becomes cumbersome. Here's how it works for x² + 4x - 21:

1. Find the product AC: In our case, a = 1 and c = -21, so AC = 1 * (-21) = -21.

2. Find two numbers that add up to B and multiply to AC: We need two numbers that add up to 4 (our b value) and multiply to -21. As we found in Method 1, these numbers are -3 and 7.

3. Rewrite the expression: We rewrite the middle term (4x) as the sum of these two numbers: x² - 3x + 7x - 21.

4. Factor by grouping: Group the terms in pairs and factor out the greatest common factor (GCF) from each pair:

   x²(x - 3) + 7(x - 3)

5. Factor out the common binomial: Notice that both terms now have (x - 3) as a common factor. Factor this out:

   (x - 3)(x + 7)

Therefore, using the AC method, we again arrive at the factored form (x - 3)(x + 7).

Method 3: Quadratic Formula (Indirect Factoring)

While not strictly a factoring method, the quadratic formula can be used to find the roots of the quadratic equation x² + 4x - 21 = 0. These roots can then be used to construct the factored form. The quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

Plugging in our values (a = 1, b = 4, c = -21), we get:

x = [-4 ± √(4² - 4 * 1 * -21)] / 2 * 1

x = [-4 ± √(16 + 84)] / 2

x = [-4 ± √100] / 2

x = [-4 ± 10] / 2

This gives us two solutions:

x = (-4 + 10) / 2 = 3

x = (-4 - 10) / 2 = -7

These are the roots of the equation. Since (x - r) is a factor if r is a root, the factored form is (x - 3)(x + 7).

Solving Quadratic Equations using the Factored Form

Once we have the factored form (x - 3)(x + 7), we can easily solve the corresponding quadratic equation x² + 4x - 21 = 0. The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore:

(x - 3) = 0 or (x + 7) = 0

Solving for x gives us the solutions: x = 3 and x = -7. These are the x-intercepts of the parabola represented by the equation y = x² + 4x - 21.

Graphical Representation

Graphing the quadratic equation y = x² + 4x - 21 visually confirms our findings. The parabola will intersect the x-axis at x = 3 and x = -7, representing the roots of the equation and confirming the accuracy of our factoring.

Expanding the Factored Form

To further verify our work, we can expand the factored form (x - 3)(x + 7) using the FOIL method (First, Outer, Inner, Last):

• First: x * x = x²
• Outer: x * 7 = 7x
• Inner: -3 * x = -3x
• Last: -3 * 7 = -21

Combining like terms, we get x² + 7x - 3x - 21 = x² + 4x - 21, which is our original quadratic expression.

Frequently Asked Questions (FAQ)

• Q: What if the quadratic expression doesn't factor easily?

  • A: If the expression doesn't factor easily using inspection or the AC method, you can use the quadratic formula to find the roots and then construct the factored form. Alternatively, you might need to use techniques like completing the square.

• Q: Can a quadratic expression have only one factor?

  • A: No, a quadratic expression will always have two factors, although they may be the same (resulting in a perfect square trinomial).

• Q: What happens if 'a' is not equal to 1?

  • A: If 'a' is not 1, the factoring process becomes slightly more complex. You might need to use the AC method more carefully, or consider other advanced factoring techniques.

• Q: How can I check if my factoring is correct?

  • A: Always expand your factored form using the FOIL method or distributive property to verify that it results in the original quadratic expression.

Conclusion

Factoring quadratic expressions is a crucial skill in algebra. We've explored three different methods—factoring by inspection, the AC method, and using the quadratic formula—demonstrating their application to the expression x² + 4x - 21. Understanding these methods allows you to solve quadratic equations, graph parabolas, and build a stronger foundation for more advanced mathematical concepts. Remember to practice regularly to build your proficiency and confidence in tackling these types of problems. The more you practice, the easier and more intuitive the process will become. Don't be discouraged if you find it challenging initially; with persistence and consistent effort, you will master this essential algebraic skill.