Quadratic Equations And Curves Determining Intersections With The X-Axis

In the captivating realm of mathematics, quadratic equations and curves hold a position of paramount significance. This article delves into the fascinating interplay between the roots of a quadratic equation and the points of intersection of a related quadratic curve with the x-axis. Specifically, we will explore the scenario where we have real numbers a, b, c, and d, with the condition that 1 is a root of the quadratic equation ax² + bx + c = 0. Our primary objective is to determine how the curve y = 4ax² + 3bx + 2c, where a ≠ 0, intersects the x-axis. This exploration involves a careful blend of algebraic manipulation and geometric interpretation, providing a rich understanding of the relationship between equations and their graphical representations. Understanding these concepts is crucial for anyone delving deeper into mathematical analysis and its applications.

Let's lay the groundwork for our investigation by revisiting the fundamental concepts of quadratic equations and their roots. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are coefficients, and a is not equal to zero. The roots of a quadratic equation are the values of x that satisfy the equation, or in geometric terms, the x-coordinates where the corresponding quadratic curve intersects the x-axis. These roots can be real or complex, and their nature is determined by the discriminant of the quadratic equation. The discriminant, denoted as Δ, is given by the formula Δ = b² - 4ac. If Δ > 0, the equation has two distinct real roots; if Δ = 0, it has one real root (a repeated root); and if Δ < 0, it has two complex roots. In our case, we are given that 1 is a root of the equation ax² + bx + c = 0. This crucial piece of information will serve as the cornerstone of our analysis. Since 1 is a root, substituting x = 1 into the equation gives us a(1)² + b(1) + c = 0, which simplifies to a + b + c = 0. This relationship between the coefficients a, b, and c will be instrumental in understanding the behavior of the curve y = 4ax² + 3bx + 2c.

Our attention now shifts to the curve y = 4ax² + 3bx + 2c. To analyze its intersection with the x-axis, we need to understand its properties as a quadratic function. The x-axis intersections occur where y = 0, so we are essentially seeking the roots of the quadratic equation 4ax² + 3bx + 2c = 0. To determine the nature and number of these roots, we will once again employ the discriminant. Let's denote the discriminant of this new quadratic equation as Δ'. Using the formula for the discriminant, we have Δ' = (3b)² - 4(4a)(2c) = 9b² - 32ac. The sign of Δ' will dictate how many times the curve intersects the x-axis. If Δ' > 0, the curve intersects the x-axis at two distinct points; if Δ' = 0, it touches the x-axis at exactly one point; and if Δ' < 0, it does not intersect the x-axis at all. The challenge now lies in relating Δ' to the given condition that 1 is a root of ax² + bx + c = 0, or equivalently, a + b + c = 0. We need to manipulate the expression for Δ' and use the relationship a + b + c = 0 to draw conclusions about its sign. This is where algebraic ingenuity comes into play, allowing us to bridge the gap between the root of one quadratic equation and the intersection behavior of another.

To proceed with our analysis, we need to express Δ' in a form that allows us to utilize the condition a + b + c = 0. From this condition, we can write b = -a - c. Now, let's substitute this expression for b into the discriminant Δ' = 9b² - 32ac: Δ' = 9(-a - c)² - 32ac Expanding the squared term, we get: Δ' = 9(a² + 2ac + c²) - 32ac Δ' = 9a² + 18ac + 9c² - 32ac Δ' = 9a² - 14ac + 9c² Our goal is to determine the sign of Δ'. To do this, we can try to rewrite Δ' as a sum of squares, which is always non-negative. This will give us a clear indication of whether Δ' is positive, negative, or zero. To rewrite Δ', we can complete the square. Let's focus on the terms involving a: Δ' = 9(a² - (14/9)ac) + 9c² To complete the square, we need to add and subtract (7/9)c inside the parenthesis: Δ' = 9(a² - (14/9)ac + (7/9)²c² - (7/9)²c²) + 9c² Δ' = 9(a - (7/9)c)² - 9(49/81)c² + 9c² Δ' = 9(a - (7/9)c)² - (49/9)c² + (81/9)c² Δ' = 9(a - (7/9)c)² + (32/9)c² Now we have expressed Δ' as the sum of two squares. Since squares of real numbers are always non-negative, we know that 9(a - (7/9)c)² ≥ 0 and (32/9)c² ≥ 0. Therefore, Δ' = 9(a - (7/9)c)² + (32/9)c² ≥ 0. The discriminant Δ' is always non-negative. We now need to consider when Δ' can be equal to zero. For Δ' to be zero, both terms in the sum must be zero: 9(a - (7/9)c)² = 0 and (32/9)c² = 0 From the second equation, we have c = 0. Substituting c = 0 into the first equation gives us 9(a)² = 0, which implies a = 0. However, we are given that a ≠ 0, so this case is not possible. Therefore, Δ' must be strictly greater than zero. Since Δ' > 0, the quadratic equation 4ax² + 3bx + 2c = 0 has two distinct real roots. This means that the curve y = 4ax² + 3bx + 2c intersects the x-axis at exactly two distinct points.

In conclusion, our meticulous analysis, combining algebraic techniques with insights from quadratic equation theory, has led us to a definitive answer. Given that 1 is a root of the quadratic equation ax² + bx + c = 0 and a ≠ 0, the curve y = 4ax² + 3bx + 2c intersects the x-axis at exactly two distinct points. This result showcases the powerful interplay between the roots of a quadratic equation and the geometric behavior of the corresponding quadratic curve. The use of the discriminant, combined with strategic algebraic manipulation, allowed us to bridge the connection between the given root and the intersection properties of the curve. This exploration not only provides a solution to the specific problem but also deepens our understanding of the fundamental relationships within quadratic functions and their graphical representations. By carefully analyzing the discriminant and employing algebraic techniques, we've illuminated the intricate relationship between equations and curves, solidifying our understanding of these core mathematical concepts. The ability to connect algebraic conditions with geometric outcomes is a testament to the elegance and interconnectedness of mathematics, providing a framework for solving complex problems with clarity and precision.