N 4 - 9 - 5n

Exploring the Quadratic Expression: 4 - 9n - 5n²

The quadratic expression 4 - 9n - 5n² represents a fundamental concept in algebra. Understanding its properties, such as finding its roots, determining its vertex, and sketching its graph, is crucial for various mathematical applications. This comprehensive guide delves into the intricacies of this expression, providing a step-by-step approach to analyzing and interpreting its characteristics. We'll cover various methods for solving, including factoring, the quadratic formula, and completing the square, along with a discussion of its graphical representation and real-world applications.

Understanding Quadratic Expressions

Before diving into the specifics of 4 - 9n - 5n², 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 (in this case, 'n') is 2. The general form of a quadratic expression is:

ax² + bx + c

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. In our specific case, 4 - 9n - 5n², we have:

• a = -5
• b = -9
• c = 4

The negative value of 'a' indicates that the parabola representing this quadratic expression will open downwards. This implies the quadratic has a maximum value, not a minimum.

Finding the Roots (or Zeros) of the Quadratic

The roots of a quadratic equation are the values of 'n' that make the expression equal to zero. Finding these roots is essential for understanding the behavior of the quadratic. There are several methods to achieve this:

1. Factoring

Factoring involves expressing the quadratic expression as a product of two linear expressions. While not all quadratics can be easily factored, this is often the quickest method when it's possible. Let's attempt to factor 4 - 9n - 5n²:

We're looking for two numbers that add up to -9 (the coefficient of 'n') and multiply to -20 (the product of 'a' and 'c', which is -5 * 4 = -20). These numbers are -15 and 4. Therefore we can rewrite and factor the expression as follows:

-5n² - 9n + 4 = -(5n² + 9n - 4) = -(5n - 1)(n + 4)

Setting this equal to zero gives us the roots:

• (5n - 1) = 0 => n = 1/5
• (n + 4) = 0 => n = -4

Thus, the roots of the quadratic expression are n = 1/5 and n = -4.

2. The Quadratic Formula

The quadratic formula is a more general method that works for all quadratic equations, regardless of their factorability. The formula is:

n = [-b ± √(b² - 4ac)] / 2a

Substituting the values from our expression:

n = [9 ± √((-9)² - 4 * (-5) * 4)] / (2 * -5) n = [9 ± √(81 + 80)] / -10 n = [9 ± √161] / -10

This gives us two roots:

• n ≈ -4
• n ≈ 0.2

These values closely approximate the roots we obtained through factoring. The slight discrepancy is due to rounding in the quadratic formula calculation.

3. Completing the Square

Completing the square is another algebraic technique to solve quadratic equations. It involves manipulating the expression to create a perfect square trinomial. This method can be useful for finding the vertex of the parabola, which we'll discuss later. While less intuitive than factoring or the quadratic formula, it provides valuable insights into the quadratic's structure.

Finding the Vertex of the Parabola

The vertex of a parabola represents its highest or lowest point. Since our quadratic opens downwards (a = -5), the vertex will represent the maximum value of the expression. The x-coordinate of the vertex can be found using the formula:

n = -b / 2a

Substituting our values:

n = -(-9) / (2 * -5) = 9 / -10 = -0.9

To find the y-coordinate (the maximum value of the expression), we substitute this value of 'n' back into the original quadratic expression:

4 - 9(-0.9) - 5(-0.9)² ≈ 8.05

Therefore, the vertex of the parabola is approximately (-0.9, 8.05).

Sketching the Graph of the Quadratic

Now that we have the roots and the vertex, we can sketch a reasonable representation of the parabola. Remember, the parabola opens downwards, crosses the n-axis at approximately -4 and 0.2, and has a maximum point at approximately (-0.9, 8.05).

(Imagine a graph here showing a downward-opening parabola with the intercepts and vertex labeled)

Real-World Applications

Quadratic expressions have numerous applications in various fields:

• Physics: Describing projectile motion (the trajectory of a ball or rocket).
• Engineering: Modeling the strength of beams or structures.
• Economics: Representing cost functions or revenue curves.
• Computer Graphics: Creating curved lines and shapes.

In each of these applications, understanding the roots, vertex, and overall shape of the quadratic is crucial for making accurate predictions and informed decisions.

Further Exploration: Discriminant and Nature of Roots

The discriminant (b² - 4ac) within the quadratic formula provides insight into the nature of the roots:

• If b² - 4ac > 0: The quadratic has two distinct real roots (as in our example).
• If b² - 4ac = 0: The quadratic has one repeated real root.
• If b² - 4ac < 0: The quadratic has two complex roots (no real solutions).

In our case, b² - 4ac = 161 > 0, confirming our finding of two distinct real roots.

Conclusion

The quadratic expression 4 - 9n - 5n² provides a rich example of the concepts within quadratic functions. By employing various methods such as factoring, the quadratic formula, and completing the square, we can thoroughly analyze its properties, including its roots, vertex, and graphical representation. This understanding extends to various real-world applications where quadratic models are essential for problem-solving and prediction. This detailed analysis emphasizes the importance of mastering quadratic expressions as a foundational element of algebra and its numerous practical applications. Remember to practice these methods with various quadratic expressions to build your proficiency and understanding of these fundamental mathematical tools.