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

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

This article will explore the quadratic expression 4x² + 19x - 5, examining its properties and determining which statements regarding its characteristics are true. We'll delve into factoring, finding roots, analyzing the parabola it represents, and understanding its behavior. This comprehensive analysis will provide a robust understanding of quadratic expressions and their applications.

Understanding Quadratic Expressions

Before we dive into the specifics of 4x² + 19x - 5, 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 is ax² + bx + c, where a, b, and c are constants and a ≠ 0. The coefficient 'a' significantly influences the parabola's shape (opening upwards or downwards), while 'b' and 'c' affect its position and intercepts.

Our target expression, 4x² + 19x - 5, fits this general form with a = 4, b = 19, and c = -5. Understanding these coefficients is crucial for analyzing its properties.

Factoring the Quadratic Expression

Factoring a quadratic expression involves rewriting it as a product of two linear expressions. This process is essential for finding the roots (x-intercepts) and simplifying expressions. There are several methods for factoring, including:

• Trial and Error: This method involves systematically testing different combinations of factors of 'a' and 'c' until the correct combination that yields the middle term 'b' is found. This can be time-consuming for complex expressions.

• Quadratic Formula: This formula directly calculates the roots of the quadratic equation ax² + bx + c = 0, and from these roots, the factored form can be derived. The formula is:

  x = [-b ± √(b² - 4ac)] / 2a

• Completing the Square: This method involves manipulating the expression to create a perfect square trinomial, which can then be easily factored.

Let's apply the quadratic formula to our expression:

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:

x₁ = (-19 + 21) / 8 = 2/8 = 1/4 x₂ = (-19 - 21) / 8 = -40/8 = -5

Therefore, the factored form of 4x² + 19x - 5 is (4x - 1)(x + 5).

Finding the Roots (x-intercepts)

The roots of a quadratic expression are the values of x that make the expression equal to zero. These are also the x-intercepts of the parabola represented by the quadratic. We've already found the roots using the quadratic formula: x = 1/4 and x = -5. These are the points where the parabola intersects the x-axis.

Analyzing the Parabola

The quadratic expression 4x² + 19x - 5 represents a parabola. Since the coefficient 'a' (which is 4) is positive, the parabola opens upwards. This means it has a minimum value (vertex).

Vertex: The x-coordinate of the vertex is given by -b/2a = -19/(2*4) = -19/8. Substituting this into the original equation gives the y-coordinate of the vertex.

Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex. Its equation is x = -b/2a = -19/8.

y-intercept: The y-intercept is the point where the parabola intersects the y-axis. This occurs when x = 0. Substituting x = 0 into the equation gives y = -5. Therefore, the y-intercept is (0, -5).

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

Now, let's consider various statements about the quadratic expression and determine their validity based on our analysis:

Statement 1: The expression can be factored. TRUE. We've successfully factored the expression into (4x - 1)(x + 5).

Statement 2: The parabola opens downwards. FALSE. The parabola opens upwards because the coefficient of x² (a = 4) is positive.

Statement 3: The roots are x = 1/4 and x = -5. TRUE. These are the solutions we obtained using the quadratic formula.

Statement 4: The y-intercept is (0, 5). FALSE. The y-intercept is (0, -5), as determined by substituting x = 0 into the equation.

Statement 5: The vertex has an x-coordinate of -19/8. TRUE. This is the x-coordinate of the vertex, as calculated using -b/2a.

Statement 6: The expression is a linear equation. FALSE. It's a quadratic expression because the highest power of x is 2.

Statement 7: The discriminant is positive. TRUE. The discriminant (b² - 4ac) is 441, which is positive. A positive discriminant indicates that the quadratic equation has two distinct real roots.

Statement 8: The expression has only one real root. FALSE. It has two distinct real roots, as evidenced by the factoring and the positive discriminant.

Statement 9: The axis of symmetry is x = -19/8. TRUE. This is the equation of the vertical line passing through the vertex.

Applications of Quadratic Expressions

Quadratic expressions have numerous applications in various fields:

• Physics: Describing projectile motion, calculating areas, and modeling certain types of oscillations.

• Engineering: Designing parabolic antennas, bridges, and other structures.

• Economics: Modeling cost functions, revenue functions, and profit maximization.

• Computer Graphics: Creating curved shapes and representing trajectories.

Conclusion

Analyzing the quadratic expression 4x² + 19x - 5 provides a rich illustration of the properties and behaviors of quadratic functions. By understanding factoring, finding roots, determining the parabola's characteristics, and evaluating statements about the expression, we've gained a deeper appreciation for this fundamental concept in algebra and its widespread applications across various disciplines. Remember that a strong grasp of quadratic expressions is crucial for success in higher-level mathematics and related fields. Continue practicing with different quadratic expressions to strengthen your understanding and problem-solving skills.