6x2 + 13x + 6

Decoding the Quadratic: A Deep Dive into 6x² + 13x + 6

This article provides a comprehensive exploration of the quadratic expression 6x² + 13x + 6, covering its factorization, graphing, solving for its roots, and the underlying mathematical concepts. Understanding this seemingly simple expression offers a gateway to comprehending more complex algebraic manipulations and problem-solving strategies in mathematics. We'll break down the process step-by-step, making it accessible to students of all levels.

Introduction: Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, meaning the highest power of the variable (in this case, x) is 2. They are often written in the standard form: ax² + bx + c, where a, b, and c are constants. Our focus, 6x² + 13x + 6, perfectly fits this mold, with a = 6, b = 13, and c = 6. Understanding quadratic expressions is crucial in various fields, from physics (calculating projectile motion) to economics (modeling supply and demand).

Method 1: Factoring the Quadratic Expression

Factoring a quadratic expression involves breaking it down into two simpler expressions that, when multiplied, give the original expression. This is a fundamental skill in algebra and allows us to solve quadratic equations and simplify more complex expressions.

There are several methods for factoring quadratics. For 6x² + 13x + 6, we'll use the "ac method."

Steps using the ac method:

1. Find the product ac: In our case, a = 6 and c = 6, so ac = 6 * 6 = 36.

2. Find two numbers that add up to b and multiply to ac: We need two numbers that add up to 13 (our b value) and multiply to 36. These numbers are 9 and 4 (9 + 4 = 13 and 9 * 4 = 36).

3. Rewrite the expression: Replace the middle term (13x) with the two numbers we found: 6x² + 9x + 4x + 6

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

   3x(2x + 3) + 2(2x + 3)

5. Factor out the common binomial: Notice that both terms now share the binomial (2x + 3). Factor this out:

   (2x + 3)(3x + 2)

Therefore, the factored form of 6x² + 13x + 6 is (2x + 3)(3x + 2). This means that if you were to multiply (2x + 3) and (3x + 2) using the FOIL method (First, Outer, Inner, Last), you would obtain the original expression.

Method 2: Using the Quadratic Formula

The quadratic formula is a powerful tool for finding the roots (solutions) of any quadratic equation of the form ax² + bx + c = 0. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Applying this to our expression, 6x² + 13x + 6 = 0, we have:

a = 6, b = 13, c = 6

x = [-13 ± √(13² - 4 * 6 * 6)] / (2 * 6) x = [-13 ± √(169 - 144)] / 12 x = [-13 ± √25] / 12 x = [-13 ± 5] / 12

This gives us two solutions:

x₁ = (-13 + 5) / 12 = -8 / 12 = -2/3 x₂ = (-13 - 5) / 12 = -18 / 12 = -3/2

These solutions correspond to the roots of the factored expression. Setting each factor to zero and solving for x gives us the same results:

2x + 3 = 0 => x = -3/2 3x + 2 = 0 => x = -2/3

Graphing the Quadratic Expression

The graph of a quadratic expression is a parabola. The parabola opens upwards if a (the coefficient of x²) is positive, and downwards if a is negative. In our case, a = 6, so the parabola opens upwards.

The x-intercepts of the parabola are the points where the graph crosses the x-axis. These intercepts correspond to the roots of the quadratic equation (which we already found to be -2/3 and -3/2). The y-intercept is the point where the graph crosses the y-axis, which occurs when x = 0. Substituting x = 0 into our expression gives us:

6(0)² + 13(0) + 6 = 6

Therefore, the y-intercept is (0, 6). The vertex (the lowest point of the parabola) can be found using the formula x = -b/2a:

x = -13 / (2 * 6) = -13/12

Substituting this x-value back into the original expression gives the y-coordinate of the vertex.

The Discriminant and Nature of Roots

The expression inside the square root in the quadratic formula (b² - 4ac) is called the discriminant. It tells us about the nature of the roots:

• b² - 4ac > 0: The quadratic equation has two distinct real roots.
• b² - 4ac = 0: The quadratic equation has one real root (a repeated root).
• b² - 4ac < 0: The quadratic equation has no real roots (the roots are complex).

In our case, b² - 4ac = 13² - 4 * 6 * 6 = 25 > 0. This confirms that our quadratic expression has two distinct real roots, which we found earlier.

Applications of Quadratic Expressions

Quadratic expressions have widespread applications across various fields:

• Physics: Calculating projectile motion, where the height of an object over time follows a parabolic path.
• Engineering: Designing structures, optimizing shapes for strength and stability.
• Economics: Modeling supply and demand curves, determining optimal pricing strategies.
• Computer Graphics: Creating curved shapes and animations.

Frequently Asked Questions (FAQ)

Q: Can I factor all quadratic expressions?

A: Not all quadratic expressions can be factored easily using integers. Some may require the use of the quadratic formula or other advanced techniques.

Q: What if the coefficient of x² is negative?

A: If the coefficient of x² is negative, you can factor out a -1 before proceeding with factoring or using the quadratic formula. This makes the calculations easier.

Q: What is the difference between a quadratic expression and a quadratic equation?

A: A quadratic expression is a polynomial of degree two, while a quadratic equation is a quadratic expression set equal to zero. Solving a quadratic equation means finding the values of x that make the equation true.

Q: How can I check if my factored expression is correct?

A: Multiply the factors back together using the FOIL method. If you get the original quadratic expression, your factorization is correct.

Conclusion: Mastering Quadratic Expressions

Understanding quadratic expressions like 6x² + 13x + 6 is fundamental to mastering algebra and its applications. Through factoring, using the quadratic formula, and graphing, we can thoroughly analyze these expressions and solve related equations. The techniques discussed here provide a strong foundation for tackling more complex mathematical problems in various fields of study and professional endeavors. Remember to practice consistently to build your proficiency and confidence in working with quadratic expressions. By understanding the underlying principles and applying the various methods, you'll find yourself equipped to confidently solve a wide range of quadratic problems.