2x 2 7x 9 0 Quadratic Formula

Decoding the Enigma: A Deep Dive into the 2x² + 7x + 9 = 0 Quadratic Equation

The seemingly simple equation, 2x² + 7x + 9 = 0, hides a wealth of mathematical concepts within its concise form. This article delves deep into solving this specific quadratic equation, exploring various methods, interpreting the results, and extending the understanding to broader applications of quadratic equations in various fields.

Understanding Quadratic Equations

Before we tackle our specific equation, let's establish a foundational understanding of 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 is represented as:

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). These constants can be any real numbers, positive, negative, or zero. The solutions to the quadratic equation, often called roots, zeros, or x-intercepts, represent the x-values where the corresponding quadratic function intersects the x-axis.

Method 1: The Quadratic Formula – A Universal Solver

The quadratic formula is a powerful tool that provides a direct solution for any quadratic equation, regardless of the values of a, b, and c. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Let's apply this to our equation, 2x² + 7x + 9 = 0. Here, a = 2, b = 7, and c = 9. Substituting these values into the quadratic formula:

x = [-7 ± √(7² - 4 * 2 * 9)] / (2 * 2) x = [-7 ± √(49 - 72)] / 4 x = [-7 ± √(-23)] / 4

Notice that we have a negative number under the square root. This signifies that the solutions to this quadratic equation are complex numbers, involving the imaginary unit 'i', where i² = -1.

Therefore, the solutions are:

x = [-7 ± i√23] / 4

This means the equation has two complex conjugate roots:

x₁ = (-7 + i√23) / 4 and x₂ = (-7 - i√23) / 4

Method 2: Completing the Square – A Geometric Approach

Completing the square is another method for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily factored. Let's apply this to our equation:

2x² + 7x + 9 = 0

1. Divide by 'a': Divide the entire equation by 2: x² + (7/2)x + (9/2) = 0

2. Move the constant term: Move the constant term (9/2) to the right side of the equation: x² + (7/2)x = -9/2

3. Complete the square: To complete the square, take half of the coefficient of x ((7/2)/2 = 7/4), square it ((7/4)² = 49/16), and add it to both sides of the equation: x² + (7/2)x + 49/16 = -9/2 + 49/16

4. Factor the perfect square trinomial: The left side is now a perfect square trinomial: (x + 7/4)² = -72/16 + 49/16 = -23/16

5. Solve for x: Take the square root of both sides: x + 7/4 = ±√(-23/16) = ±(i√23)/4

6. Isolate x: x = -7/4 ± (i√23)/4

This yields the same complex conjugate roots as the quadratic formula.

Method 3: Factoring – A Simpler (But Not Always Possible) Approach

Factoring involves expressing the quadratic equation as a product of two linear expressions. This method is only applicable if the quadratic equation has real rational roots. In our case, since the roots are complex, factoring directly is not possible with real numbers. We would need to use complex numbers to factor it, which is less straightforward than the other methods.

Interpreting the Complex Roots

The fact that our equation yields complex roots has a significant geometric interpretation. The parabola represented by the quadratic function y = 2x² + 7x + 9 does not intersect the x-axis. The complex roots represent the points where the parabola would intersect the x-axis if we extended the x-axis into the complex plane.

Applications of Quadratic Equations

Quadratic equations have widespread applications across various fields:

• Physics: Calculating projectile motion, determining the trajectory of objects under the influence of gravity.
• Engineering: Designing parabolic antennas, bridges, and other structures.
• Economics: Modeling supply and demand, optimizing production processes.
• Computer Graphics: Creating curves and shapes.
• Finance: Calculating compound interest and investment growth.

Extending the Understanding: Discriminant and Nature of Roots

The expression b² - 4ac, which appears under the square root in the quadratic formula, is called the discriminant. The discriminant determines the nature of the roots:

• b² - 4ac > 0: Two distinct real roots. The parabola intersects the x-axis at two different points.
• b² - 4ac = 0: One real root (a repeated root). The parabola touches the x-axis at one point (the vertex).
• b² - 4ac < 0: Two complex conjugate roots. The parabola does not intersect the x-axis.

In our case, the discriminant is 49 - 72 = -23, which is less than 0, confirming the presence of two complex conjugate roots.

Conclusion: Beyond the Numbers

Solving 2x² + 7x + 9 = 0 is more than just a mathematical exercise. It's a journey into the world of quadratic equations, revealing the power and elegance of mathematical tools like the quadratic formula and completing the square. Understanding the nature of roots, the significance of the discriminant, and the broader applications of quadratic equations provides a deeper appreciation for the fundamental role these equations play in various aspects of science, engineering, and beyond. This seemingly simple equation opens doors to a wealth of mathematical knowledge and practical applications. The exploration continues, extending beyond the solution itself to encompass the rich tapestry of mathematical principles it embodies.