Solve 4x 2 25 0

Solving the Quadratic Equation: 4x² + 25 = 0

This article will explore the solution of the quadratic equation 4x² + 25 = 0. We'll delve into the steps involved, the underlying mathematical concepts, and address frequently asked questions. Understanding how to solve this seemingly simple equation provides a strong foundation for tackling more complex quadratic problems. This guide is designed for students and anyone seeking to strengthen their understanding of quadratic equations and complex numbers.

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 ≠ 0. Our equation, 4x² + 25 = 0, is a special case where b = 0.

Steps to Solve 4x² + 25 = 0

Let's solve the equation step-by-step:

1. Isolate the x² term: Subtract 25 from both sides of the equation:

   4x² = -25

2. Solve for x²: Divide both sides by 4:

   x² = -25/4

3. Solve for x: Take the square root of both sides. Remember that the square root of a negative number involves imaginary numbers:

   x = ±√(-25/4)

4. Simplify: We can simplify this expression. Recall that √(-1) is defined as the imaginary unit i. Therefore:

   x = ±(√25/√4) * √(-1) x = ±(5/2)*i

5. Final Solution: The solutions to the equation 4x² + 25 = 0 are:

   x = (5/2)i and x = -(5/2)i

These solutions are complex numbers, each consisting of a real part (0) and an imaginary part (5/2 and -5/2 respectively).

A Deeper Dive into the Mathematics

The solution highlights the concept of complex numbers. Real numbers are numbers that can be plotted on a number line. Complex numbers extend this concept by including an imaginary unit, i, where i² = -1. A complex number is expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. In our solution, 'a' is 0, and 'b' is 5/2 or -5/2.

The fact that we obtained imaginary solutions indicates that the parabola represented by the equation 4x² + 25 = 0 does not intersect the x-axis. If you were to graph this quadratic equation, you would see a parabola that opens upwards but lies entirely above the x-axis. This is because the parabola's vertex is at (0, 25), and it never touches or crosses the x-axis where y = 0.

Alternative Methods for Solving Quadratic Equations

While the above method is straightforward for this particular equation, other methods can be used to solve quadratic equations in general. Let's briefly review some:

• Factoring: This method involves expressing the quadratic equation as a product of two linear factors. It's most effective when the equation can be easily factored. Our equation, however, cannot be factored using real numbers.

• Quadratic Formula: The quadratic formula is a powerful tool that can be used to solve any quadratic equation, even those with complex solutions. The formula is:

  x = [-b ± √(b² - 4ac)] / 2a

  In our case, a = 4, b = 0, and c = 25. Substituting these values into the formula will yield the same solutions we obtained earlier: (5/2)i and -(5/2)i. The term inside the square root, b² - 4ac, is called the discriminant. A negative discriminant indicates complex solutions.

• Completing the Square: This method involves manipulating the equation to form a perfect square trinomial, which can then be easily factored. While applicable to this problem, it's less efficient than the direct method used above.

Visualizing the Solution

While we can't directly plot complex numbers on a standard Cartesian coordinate system (x-y plane), we can visualize the quadratic equation and understand why the solutions are complex. The graph of y = 4x² + 25 is a parabola that opens upwards and has its vertex at (0, 25). Since the parabola lies entirely above the x-axis, there are no real number solutions where the curve intersects the x-axis (y=0). The complex solutions represent the points where the parabola would intersect the x-axis if we were to extend the graph into the complex plane. This requires a more advanced mathematical concept, often visualized using a complex plane with a real axis and an imaginary axis.

Frequently Asked Questions (FAQ)

Q: What are imaginary numbers used for?

A: Imaginary numbers might seem abstract, but they have crucial applications in various fields like:

• Electrical Engineering: Analyzing alternating current circuits.
• Quantum Mechanics: Describing the behavior of subatomic particles.
• Signal Processing: Modeling waves and oscillations.
• Fluid Dynamics: Solving complex flow problems.

Q: Can a quadratic equation have only one solution?

A: Yes, a quadratic equation can have one real solution (when the discriminant is 0) or two distinct real solutions (when the discriminant is positive). In our case, with a negative discriminant, we have two distinct complex solutions.

Q: Why is the solution expressed with 'i'?

A: The symbol 'i' represents the imaginary unit, defined as the square root of -1 (i = √-1). It's a necessary component for working with complex numbers, allowing us to represent the square roots of negative numbers.

Q: What does the ± symbol mean?

A: The ± symbol (plus-minus) indicates that there are two possible solutions: one with a positive value and one with a negative value. This arises from taking the square root, as both positive and negative numbers, when squared, result in a positive number.

Q: How can I check my solution?

A: To verify your solution, substitute the values of x ((5/2)i and -(5/2)i) back into the original equation (4x² + 25 = 0). You should find that the equation holds true in both cases, demonstrating that these are indeed the solutions. Remember to use the property that i² = -1 during the substitution and simplification.

Conclusion

Solving 4x² + 25 = 0 leads us to explore the fascinating world of complex numbers. While the equation initially appears simple, its solution reveals deeper mathematical concepts that are fundamental to various scientific and engineering disciplines. Understanding this solution builds a strong foundation for tackling more challenging quadratic equations and provides a deeper appreciation for the intricacies of algebra and the number systems. Mastering quadratic equations is a vital skill in mathematics, serving as a stepping stone to more advanced mathematical concepts. Through understanding the methods and the underlying principles, we can confidently approach and solve a wide range of quadratic equations.