In the realm of quadratic equations, the nature of the roots plays a pivotal role in determining the solutions. The roots can either be real or complex, and understanding this distinction is crucial for various mathematical and real-world applications. In this section, we will explore the intricacies of the discriminant analysis and the fascinating world of complex solutions.
Discriminant Analysis
At the heart of understanding the nature of the roots of a quadratic equation lies the discriminant. It's a simple yet powerful tool that provides immediate insights into the type of roots a quadratic equation possesses.
What is the Discriminant?
The discriminant of a quadratic equation is a part of the quadratic formula and is denoted by the symbol 'D'. It is calculated using the coefficients of the quadratic equation in the form ax2 + bx + c = 0. The formula for the discriminant is:
Discriminant (D) = b2 - 4ac
Interpreting the Discriminant
The value of the discriminant can tell us the nature of the roots of the quadratic equation:
- D > 0: If the discriminant is positive, the quadratic equation has two distinct real roots. This means the parabola represented by the equation intersects the x-axis at two distinct points.
- D = 0: A discriminant value of zero indicates that the quadratic equation has one repeated real root. Graphically, this means the parabola touches the x-axis at a single point, known as the vertex.
- D < 0: A negative discriminant reveals that the quadratic equation has no real roots but two complex conjugate roots. In terms of the graph, the parabola does not intersect the x-axis at all.
Complex Solutions
When the discriminant is negative, it indicates that the quadratic equation does not have real roots. Instead, it has complex roots. These roots are a combination of real and imaginary parts and are represented in the form a ± bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit defined by the equation i^2 = -1.
Understanding Complex Numbers
Complex numbers might seem abstract at first, but they are fundamental in various mathematical and scientific domains. A complex number has two parts: the real part (a) and the imaginary part (bi). The 'i' in the imaginary part represents the square root of -1.
Finding Complex Roots
To determine the complex roots of a quadratic equation, we employ the quadratic formula:
x = (-b ± sqrt(D)) / 2a
Given that the discriminant is negative, the square root of D will be imaginary, leading to complex solutions.
Real-world Implications of Complex Solutions
While complex numbers might seem distant from everyday experiences, they are deeply embedded in many scientific and engineering problems. For instance:
- Electrical Engineering: Complex numbers are indispensable when analysing alternating current (AC) circuits. They help engineers understand the phase difference between voltage and current.
- Fluid Dynamics: In the study of fluid flow, complex numbers assist in modelling potential flow in two-dimensional scenarios.
- Vibrations and Waves: Complex numbers aid in understanding oscillations, wave propagation, and the behaviour of vibrating systems.
Example Problems
To solidify our understanding, let's delve into some example problems:
Example 1: Determine the nature of the roots of the equation 3x2 - 12x + 13 = 0.
Using the discriminant formula, we find D = (-12)2 - 4(3)(13) = 144 - 156 = -12. Since D < 0, the equation has two complex conjugate roots.
Example 2: Solve the equation 2x2 + 4x + 17 = 0 for its roots.
First, we calculate the discriminant: D = 42 - 4(2)(17) = 16 - 136 = -120. The negative value of D indicates complex roots. Using the quadratic formula, we can determine the complex roots of the equation which are x = (-2 - i√30) / 2 and x = (-2 + i√30) / 2.
Example 3: A company's profit, based on the price of a product, is modelled by the equation p2 + 8p + 20 = 0, where p represents the price increase in percentage. Analyse the roots to interpret the price increase.
Calculating the discriminant, D = 82 - 4(1)(20) = 64 - 80 = -16. The negative value of D suggests that the profit model has complex solutions, implying that there isn't a straightforward real-world price increase that would maximise the profit based on this model.
FAQ
The graph of a quadratic equation, represented by a parabola, provides a visual representation of its roots. If the discriminant is positive, the parabola intersects the x-axis at two distinct points, representing the two real roots. If the discriminant is zero, the parabola touches the x-axis at a single point, indicating a repeated real root. If the discriminant is negative, the parabola doesn't intersect the x-axis at all, signifying that the equation has complex roots.
No, a quadratic equation cannot have one real root and one complex root. The roots of a quadratic equation with real coefficients will always either be both real or both complex conjugates of each other. This is due to the nature of polynomial equations and the fact that complex roots always appear in conjugate pairs when the coefficients of the polynomial are real.
Complex roots of quadratic equations always appear in conjugate pairs due to the symmetric nature of the equation. When we expand the product (x - (a + bi))(x - (a - bi)), where a and b are real numbers, the imaginary parts cancel out, resulting in a quadratic equation with real coefficients. Since the coefficients are real, any imaginary term in one root must be cancelled out by an opposite imaginary term in the other root, making them conjugates of each other.
The discriminant of a quadratic equation, calculated as b2 - 4ac, plays a crucial role in determining the nature of the roots of the equation. If the discriminant is positive, the quadratic equation has two distinct real roots. If it's zero, the equation has one real root (sometimes referred to as a repeated or double root). If the discriminant is negative, the equation doesn't have real roots but instead has two complex conjugate roots. Thus, without solving the equation, the discriminant gives us a quick insight into the type of solutions we can expect.
Geometrically, when a quadratic equation has complex roots, its graph (a parabola) does not intersect the x-axis. Instead, it lies entirely above or below the x-axis. While real roots can be visualised as the points where the parabola intersects the x-axis, complex roots don't have a direct geometric representation on the real number plane. Instead, they can be visualised in the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part of the complex number.
Practice Questions
To determine the nature of the roots of the quadratic equation, we need to calculate the discriminant, which is given by b2 - 4ac. For the given equation, a = 1, b = 4, and c = 8. Plugging these values in, we get: Discriminant = 42 - 4(1)(8) = 16 - 32 = -16. Since the discriminant is negative, the quadratic equation has complex roots.
To find the roots, we use the quadratic formula: x = (-b ± sqrt(b2 - 4ac)) / 2a Using the values of a, b, and c, we get: x = (-4 ± sqrt(-16)) / 2 x = -2 ± 2i So, the roots are x = -2 + 2i and x = -2 - 2i.
Given the roots 3 + 2i and 3 - 2i, we can use the fact that if p and q are the roots of a quadratic equation, then the equation can be written as: (x - p)(x - q) = 0 Substituting the given roots, we get: (x - (3 + 2i))(x - (3 - 2i)) = 0 Expanding and simplifying, we get: x2 - 6x + 13 = 0 Thus, the quadratic equation with roots 3 + 2i and 3 - 2i is x2 - 6x + 13 = 0.
