Which Statement About 4x2+19x-5 Is True

Which Statement About 4x² + 19x - 5 is True? A Deep Dive into Quadratic Equations

This article explores the quadratic equation 4x² + 19x - 5, examining various statements about its properties and determining their truthfulness. We'll delve into factoring, solving for x, finding the vertex, and analyzing the discriminant to fully understand this specific equation and its implications within the broader context of quadratic functions.

Understanding Quadratic Equations

Before we tackle the specific equation 4x² + 19x - 5, let's refresh our understanding of quadratic equations in general. A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The solutions to this equation, also known as the roots or zeros, represent the x-intercepts of the parabola represented by the equation when graphed.

The key characteristics of a quadratic equation include:

• The Parabola: When graphed, a quadratic equation always forms a parabola, a U-shaped curve. The parabola opens upwards if 'a' is positive and downwards if 'a' is negative.
• The Vertex: The vertex is the highest or lowest point on the parabola, depending on whether it opens downwards or upwards, respectively. It represents the minimum or maximum value of the function.
• The Roots (or Zeros): These are the points where the parabola intersects the x-axis. A quadratic equation can have two real roots, one real root (a repeated root), or two complex roots.
• The Discriminant: The discriminant (b² - 4ac) is a crucial part of the quadratic formula and helps determine the nature of the roots. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one real root (a repeated root), and a negative discriminant indicates two complex roots.

Factoring 4x² + 19x - 5

One common method for solving quadratic equations is factoring. Factoring involves expressing the quadratic expression as a product of two linear expressions. Let's attempt to factor 4x² + 19x - 5:

We are looking for two numbers that multiply to (4)(-5) = -20 and add up to 19. These numbers are 20 and -1. We can rewrite the equation as:

4x² + 20x - x - 5 = 0

Now, we can factor by grouping:

4x(x + 5) - 1(x + 5) = 0

(4x - 1)(x + 5) = 0

This factored form allows us to easily find the roots:

4x - 1 = 0 => x = 1/4 x + 5 = 0 => x = -5

Therefore, the roots of the equation 4x² + 19x - 5 = 0 are x = 1/4 and x = -5.

Solving Using the Quadratic Formula

The quadratic formula provides a general solution for any quadratic equation:

x = (-b ± √(b² - 4ac)) / 2a

For our equation, 4x² + 19x - 5 = 0, we have a = 4, b = 19, and c = -5. Substituting these values into the quadratic formula:

x = (-19 ± √(19² - 4 * 4 * -5)) / (2 * 4) x = (-19 ± √(361 + 80)) / 8 x = (-19 ± √441) / 8 x = (-19 ± 21) / 8

This gives us two solutions:

x = (-19 + 21) / 8 = 2/8 = 1/4 x = (-19 - 21) / 8 = -40/8 = -5

These results confirm the roots we found through factoring.

Analyzing the Discriminant

The discriminant, b² - 4ac, provides valuable information about the nature of the roots. For our equation:

b² - 4ac = 19² - 4 * 4 * -5 = 361 + 80 = 441

Since the discriminant is positive (441), the equation has two distinct real roots, which aligns with our findings.

Finding the Vertex

The x-coordinate of the vertex of a parabola is given by:

x = -b / 2a

For our equation:

x = -19 / (2 * 4) = -19/8

To find the y-coordinate, substitute this x-value back into the original equation:

y = 4(-19/8)² + 19(-19/8) - 5 y = 4(361/64) - 361/8 - 5 y = 361/16 - 722/16 - 80/16 y = -441/16

Therefore, the vertex of the parabola is (-19/8, -441/16).

Statements About 4x² + 19x - 5: Truth or False?

Now, let's evaluate several statements about the equation 4x² + 19x - 5 = 0 and determine their validity based on our analysis:

Statement 1: The equation has two real roots. TRUE. We've demonstrated this through factoring, the quadratic formula, and the positive discriminant.

Statement 2: One root is 1/4 and the other is -5. TRUE. This is directly confirmed by our calculations.

Statement 3: The parabola opens upwards. TRUE. The coefficient of the x² term (a = 4) is positive, indicating an upward-opening parabola.

Statement 4: The vertex of the parabola is at x = -19/8. TRUE. Our calculation confirms this x-coordinate of the vertex.

Statement 5: The y-intercept is -5. TRUE. The y-intercept occurs when x = 0. Substituting x = 0 into the equation gives y = -5.

Statement 6: The equation can be factored as (4x - 1)(x + 5) = 0. TRUE. We successfully factored the equation into this form.

Statement 7: The discriminant is negative. FALSE. The discriminant is 441, which is positive.

Statement 8: The equation has one repeated root. FALSE. The equation has two distinct real roots.

Statement 9: The parabola opens downwards. FALSE. The parabola opens upwards because 'a' is positive.

Conclusion

By systematically analyzing the quadratic equation 4x² + 19x - 5 = 0, employing various methods including factoring, the quadratic formula, and discriminant analysis, we have comprehensively determined the truthfulness of several statements regarding its properties. Understanding these techniques is crucial for solving quadratic equations and interpreting their graphical representations. This deeper understanding extends beyond just finding solutions; it allows us to visualize the parabola, identify its key features (vertex, intercepts), and accurately predict the nature of its roots based on the discriminant. This knowledge is fundamental in various fields, from physics and engineering to economics and computer science, where quadratic models are frequently used to describe and predict real-world phenomena.