X2 3x 10 0 Solution

Solving the Quadratic Equation: x² + 3x + 10 = 0

This article explores the solution to the quadratic equation x² + 3x + 10 = 0, delving into the various methods for solving such equations and the interpretation of the results. Understanding quadratic equations is fundamental to various fields, including mathematics, physics, engineering, and computer science. We'll not only find the solutions but also examine the nature of these solutions and their significance. This comprehensive guide will provide a clear and thorough understanding of the process.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually x) is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero (otherwise, it wouldn't be a quadratic equation). Our specific equation, x² + 3x + 10 = 0, fits this form with a = 1, b = 3, and c = 10.

Methods for Solving Quadratic Equations

Several methods exist for solving quadratic equations. The most common include:

• Factoring: This method involves rewriting the equation as a product of two linear expressions. However, not all quadratic equations can be easily factored. Our equation, x² + 3x + 10 = 0, is not easily factorable using real numbers.

• Quadratic Formula: The quadratic formula is a universal method that works for all quadratic equations, regardless of whether they are factorable or not. The formula is:

  x = (-b ± √(b² - 4ac)) / 2a

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

Applying the Quadratic Formula to x² + 3x + 10 = 0

Let's use the quadratic formula to solve our equation: x² + 3x + 10 = 0. Substituting the values a = 1, b = 3, and c = 10 into the formula, we get:

x = (-3 ± √(3² - 4 * 1 * 10)) / (2 * 1) x = (-3 ± √(9 - 40)) / 2 x = (-3 ± √(-31)) / 2

Notice that we have a negative number under the square root. The square root of a negative number is not a real number; instead, it's an imaginary number. We use the imaginary unit i, where i² = -1.

Therefore, the solutions are:

x = (-3 + i√31) / 2 and x = (-3 - i√31) / 2

These solutions are complex conjugates, meaning they have the same real part (-3/2) but opposite imaginary parts.

Understanding Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part. They are expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. In our case, the solutions are complex numbers. The presence of complex solutions indicates that the parabola represented by the quadratic equation does not intersect the x-axis (where y=0).

Graphical Representation

Graphing the quadratic equation y = x² + 3x + 10 helps visualize the solution. The parabola opens upwards (since a > 0), and its vertex lies above the x-axis. This visually confirms that there are no real roots; the parabola never intersects the x-axis. The x-intercepts (where the graph crosses the x-axis) would represent the real solutions, but in this case, they exist in the complex plane.

Completing the Square Method for x² + 3x + 10 = 0

While the quadratic formula provides a direct solution, let's also demonstrate the completing the square method.

1. Move the constant term to the right side: x² + 3x = -10

2. Take half of the coefficient of x (which is 3), square it ((3/2)² = 9/4), and add it to both sides: x² + 3x + 9/4 = -10 + 9/4

3. Simplify the right side: x² + 3x + 9/4 = -31/4

4. Factor the left side as a perfect square: (x + 3/2)² = -31/4

5. Take the square root of both sides: x + 3/2 = ±√(-31/4) = ±(i√31)/2

6. Solve for x: x = -3/2 ± (i√31)/2

This yields the same complex solutions as the quadratic formula.

Discriminant and the Nature of Roots

The expression b² - 4ac within the quadratic formula is called the discriminant. It determines the nature of the roots:

• If b² - 4ac > 0: The equation has two distinct real roots. The parabola intersects the x-axis at two points.

• If b² - 4ac = 0: The equation has one real root (a repeated root). The parabola touches the x-axis at one point (the vertex).

• If b² - 4ac < 0: The equation has two complex conjugate roots (as in our example). The parabola does not intersect the x-axis.

In our case, the discriminant is 3² - 4(1)(10) = -31, which is less than 0, indicating two complex conjugate roots.

Applications of Complex Numbers

While complex numbers might seem abstract, they have numerous practical applications in various fields:

• Electrical Engineering: Complex numbers are used extensively in analyzing alternating current (AC) circuits, where they represent impedance and phasors.

• Quantum Mechanics: Complex numbers are fundamental to the mathematical framework of quantum mechanics, describing wave functions and probabilities.

• Signal Processing: Complex numbers are crucial in analyzing and manipulating signals, such as audio and radio waves.

• Fluid Dynamics: Complex analysis is used in solving certain problems in fluid flow and aerodynamics.

Frequently Asked Questions (FAQ)

Q: Why are complex solutions important if they don't represent points on the x-axis?

A: While they don't represent points on the real number line x-axis, complex solutions are crucial in understanding the behavior of the quadratic function in the complex plane. They provide a complete picture of the function's properties.

Q: Can I solve this equation using factoring?

A: No, this particular equation cannot be factored using real numbers because its discriminant is negative. Factoring only works for equations with real roots.

Q: What does it mean for the solutions to be "conjugates"?

A: Complex conjugate pairs always appear together when solving quadratic equations with a negative discriminant. They have the same real part but opposite imaginary parts. This property is vital in various mathematical and engineering applications.

Q: Is there any other method besides the quadratic formula and completing the square?

A: While less commonly used, numerical methods like the Newton-Raphson method can also be applied to find approximate solutions to quadratic equations, including those with complex roots. However, for quadratic equations, the quadratic formula is the most efficient and straightforward method.

Conclusion

The quadratic equation x² + 3x + 10 = 0 has no real solutions. Using the quadratic formula or completing the square, we find two complex conjugate solutions: x = (-3 + i√31) / 2 and x = (-3 - i√31) / 2. The negative discriminant confirms the existence of these complex roots. Understanding the concept of complex numbers and their application is crucial for advanced studies in mathematics and various scientific and engineering disciplines. This seemingly simple equation opens doors to a richer understanding of mathematical concepts and their real-world implications.