Which Statement About 6x2 7x 10 Is True

Which Statement About 6x² + 7x + 10 is True? A Deep Dive into Quadratic Expressions

The seemingly simple expression 6x² + 7x + 10 presents a rich opportunity to explore several key concepts in algebra, particularly those related to quadratic equations. This article will delve into various statements that could be made about this expression, analyzing their truthfulness and exploring the underlying mathematical principles. We’ll cover factoring, the discriminant, the nature of roots, and graphing, providing a comprehensive understanding of this seemingly straightforward quadratic.

Understanding Quadratic Expressions

Before we tackle specific statements, let's establish a foundational understanding of quadratic expressions. A quadratic expression is an algebraic expression of the form ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. Our expression, 6x² + 7x + 10, perfectly fits this mold, with a = 6, b = 7, and c = 10.

The key characteristics of a quadratic expression are:

• The Parabola: When graphed, a quadratic expression always forms a parabola, a symmetrical U-shaped curve. The shape and position of this parabola are determined by the values of a, b, and c.
• Roots (or Zeros): These are the x-values where the parabola intersects the x-axis (where y = 0). They represent the solutions to the quadratic equation ax² + bx + c = 0.
• Vertex: This is the turning point of the parabola – the lowest point if the parabola opens upwards (a > 0), and the highest point if it opens downwards (a < 0).
• Axis of Symmetry: A vertical line that divides the parabola into two mirror-image halves. It passes through the vertex.

Analyzing Statements about 6x² + 7x + 10

Now, let's consider several statements about 6x² + 7x + 10 and determine their truthfulness. We'll use a variety of mathematical techniques to justify our conclusions.

Statement 1: "The expression 6x² + 7x + 10 can be easily factored using integers."

False. While many quadratic expressions can be factored into two simpler binomial expressions, this one cannot be factored neatly using only integers. Let's explore why. We're looking for two numbers that multiply to (6 * 10) = 60 and add up to 7. No such integer pair exists. Attempting to factor this expression would lead to the use of irrational or complex numbers.

Statement 2: "The expression 6x² + 7x + 10 has real roots."

False. To determine the nature of the roots, we can use the discriminant, denoted by Δ (Delta). The discriminant is calculated using the formula: Δ = b² - 4ac. For our expression:

Δ = (7)² - 4 * 6 * 10 = 49 - 240 = -191

Since the discriminant is negative, the quadratic equation 6x² + 7x + 10 = 0 has no real roots. The roots are complex conjugates (involving the imaginary unit 'i').

Statement 3: "The parabola represented by y = 6x² + 7x + 10 opens upwards."

True. The parabola opens upwards because the coefficient of the x² term (a) is positive (a = 6 > 0). A positive 'a' value always results in a parabola that opens upwards, indicating a minimum value at its vertex.

Statement 4: "The vertex of the parabola y = 6x² + 7x + 10 lies above the x-axis."

True. Since the parabola opens upwards and has no real roots (meaning it doesn't intersect the x-axis), its vertex must lie above the x-axis. The y-coordinate of the vertex is always positive in this scenario. We can find the x-coordinate of the vertex using the formula x = -b / 2a = -7 / (2 * 6) = -7/12. Substituting this back into the equation gives us the y-coordinate of the vertex, which will be a positive value.

Statement 5: "The y-intercept of the graph of y = 6x² + 7x + 10 is (0, 10)."

True. The y-intercept is the point where the graph intersects the y-axis (where x = 0). Substituting x = 0 into the equation gives y = 6(0)² + 7(0) + 10 = 10. Therefore, the y-intercept is indeed (0, 10).

Statement 6: "The equation 6x² + 7x + 10 = 0 has two distinct complex roots."

True. As we established earlier, the discriminant is negative (-191). A negative discriminant indicates that the quadratic equation has two distinct complex roots, which are complex conjugates of each other. These roots involve the imaginary unit 'i' (√-1).

Statement 7: "The axis of symmetry of the parabola y = 6x² + 7x + 10 is x = -7/12."

True. The axis of symmetry of a parabola given by the equation ax² + bx + c is always given by the vertical line x = -b / 2a. In our case, this is x = -7 / (2 * 6) = -7/12.

Statement 8: "The expression 6x² + 7x + 10 can be solved using the quadratic formula."

True. The quadratic formula provides a general solution for any quadratic equation of the form ax² + bx + c = 0. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Substituting the values from our expression will yield the two complex roots mentioned earlier. This confirms that the quadratic formula is applicable, even if the roots are not real.

Conclusion: Working with Quadratic Expressions

Analyzing statements about the quadratic expression 6x² + 7x + 10 has provided a thorough exploration of key concepts related to quadratic equations. We’ve seen how the discriminant determines the nature of the roots, how the coefficient of the x² term dictates the parabola's orientation, and how the quadratic formula offers a universal method for finding solutions, even when those solutions are complex. Understanding these concepts is crucial for successfully tackling more complex algebraic problems and for developing a strong foundational understanding of mathematics. Remember that practice is key – the more you work with quadratic expressions and apply these techniques, the more proficient you will become.