Factor X 2 12x 20

Factoring the Quadratic Expression: x² + 12x + 20

This article delves into the process of factoring the quadratic expression x² + 12x + 20. We'll explore different methods, explain the underlying mathematical principles, and provide a step-by-step guide to help you understand and solve similar problems. Understanding quadratic factoring is crucial for various mathematical applications, from solving equations to graphing parabolas. By the end of this article, you'll not only be able to factor this specific expression but also confidently tackle other quadratic expressions.

Understanding Quadratic Expressions

Before diving into the factoring process, 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 (usually 'x') is 2. It generally takes the form ax² + bx + c, where 'a', 'b', and 'c' are constants. In our case, a = 1, b = 12, and c = 20.

Method 1: Factoring by Inspection (Trial and Error)

This method involves finding two numbers that add up to 'b' (the coefficient of x) and multiply to 'c' (the constant term). Let's apply this to x² + 12x + 20:

1. Identify 'b' and 'c': In our expression, b = 12 and c = 20.

2. Find two numbers: We need to find two numbers that add up to 12 and multiply to 20. Let's list the factor pairs of 20:

   • 1 and 20 (sum = 21)
   • 2 and 10 (sum = 12)
   • 4 and 5 (sum = 9)

3. Use the pair that satisfies the conditions: The pair 2 and 10 satisfies both conditions (2 + 10 = 12 and 2 * 10 = 20).

4. Write the factored form: Using these numbers, we can rewrite the expression as (x + 2)(x + 10).

Therefore, the factored form of x² + 12x + 20 is (x + 2)(x + 10). You can verify this by expanding the factored form using the FOIL method (First, Outer, Inner, Last): (x + 2)(x + 10) = x² + 10x + 2x + 20 = x² + 12x + 20.

Method 2: Completing the Square

This method is more general and can be used to factor any quadratic expression, even those that are not easily factored by inspection. The process involves manipulating the expression to create a perfect square trinomial.

1. Move the constant term to the right side: Rewrite the expression as x² + 12x = -20.

2. Find the value to complete the square: Take half of the coefficient of x (12/2 = 6) and square it (6² = 36). This is the value we need to add to both sides of the equation to complete the square.

3. Add the value to both sides: x² + 12x + 36 = -20 + 36

4. Simplify: x² + 12x + 36 = 16

5. Rewrite as a perfect square: The left side is now a perfect square trinomial, which can be factored as (x + 6)².

6. Solve for x (if needed): (x + 6)² = 16 => x + 6 = ±√16 => x + 6 = ±4 => x = -2 or x = -10

7. Write the factored form: Since x = -2 and x = -10 are the roots, the factors are (x + 2) and (x + 10). Therefore, the factored form is (x + 2)(x + 10).

Method 3: Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation, which can then be used to determine the factored form. The quadratic formula is given by:

x = [-b ± √(b² - 4ac)] / 2a

where a, b, and c are the coefficients of the quadratic expression ax² + bx + c.

For our expression x² + 12x + 20 (a = 1, b = 12, c = 20):

x = [-12 ± √(12² - 4 * 1 * 20)] / (2 * 1) x = [-12 ± √(144 - 80)] / 2 x = [-12 ± √64] / 2 x = [-12 ± 8] / 2

This gives us two solutions:

x₁ = (-12 + 8) / 2 = -2 x₂ = (-12 - 8) / 2 = -10

Since x₁ = -2 and x₂ = -10 are the roots, the factors are (x + 2) and (x + 10). Therefore, the factored form is (x + 2)(x + 10).

Graphical Representation

The factored form allows us to easily determine the x-intercepts (roots) of the quadratic equation y = x² + 12x + 20. The x-intercepts are the points where the parabola intersects the x-axis (where y = 0). In our case, the x-intercepts are x = -2 and x = -10. This graphical representation visually confirms our factored form. The parabola opens upwards because the coefficient of x² is positive.

Applications of Factoring Quadratic Expressions

Factoring quadratic expressions is a fundamental skill in algebra with numerous applications, including:

• Solving quadratic equations: Setting the quadratic expression equal to zero allows you to find the roots or solutions of the equation.

• Graphing parabolas: The factored form reveals the x-intercepts, which are essential for accurately sketching the graph of a parabola.

• Simplifying algebraic expressions: Factoring can simplify complex expressions, making them easier to manipulate and solve.

• Calculus: Factoring is used extensively in calculus, particularly in finding derivatives and integrals.

• Physics and Engineering: Quadratic equations model many real-world phenomena in physics and engineering, such as projectile motion and the behavior of electrical circuits.

Frequently Asked Questions (FAQ)

• What if the quadratic expression cannot be easily factored? If the expression doesn't factor easily by inspection, you can use the quadratic formula or completing the square method.

• Are there other methods for factoring quadratic expressions? While inspection, completing the square, and the quadratic formula are the most common, other methods exist, but they are often variations or combinations of these core techniques.

• What if 'a' is not equal to 1? If the coefficient of x² (a) is not 1, the factoring process becomes slightly more complex, but the underlying principles remain the same. You might need to use techniques like factoring by grouping or employing the AC method.

• How can I check if my factoring is correct? Always expand the factored form using the FOIL method or distributive property to verify that it matches the original quadratic expression.

Conclusion

Factoring the quadratic expression x² + 12x + 20 yields the factored form (x + 2)(x + 10). We explored three different methods—factoring by inspection, completing the square, and using the quadratic formula—to demonstrate the versatility of approaches available for solving such problems. Understanding these methods is crucial not only for solving specific quadratic expressions but also for building a solid foundation in algebra and its broader applications in various fields. Remember to practice regularly to improve your proficiency in factoring quadratic expressions. The more you practice, the easier it will become to recognize patterns and efficiently apply the most suitable method for each problem. This skill will serve you well throughout your mathematical journey.