4x 2 5x 12 0 Quadratic Equations

Decoding the Enigma: A Deep Dive into 4x² + 5x - 12 = 0 Quadratic Equations

Quadratic equations, those elegant mathematical expressions of the form ax² + bx + c = 0, are fundamental to algebra and have far-reaching applications in various fields, from physics and engineering to finance and computer science. Understanding how to solve them is crucial for anyone pursuing a deeper understanding of mathematics. This comprehensive guide will delve into the specifics of solving the quadratic equation 4x² + 5x - 12 = 0, exploring multiple methods and providing a thorough understanding of the underlying principles.

Understanding the Quadratic Equation

Before we tackle the specific equation 4x² + 5x - 12 = 0, let's establish a foundational understanding of quadratic equations in general. The standard form is represented as:

ax² + bx + c = 0

Where:

• a, b, and c are constants (numbers).
• a ≠ 0 (if a were 0, the equation would become linear, not quadratic).
• x is the variable we aim to solve for.

The solutions to a quadratic equation, also known as its roots or zeros, represent the values of x that make the equation true. A quadratic equation can have two, one (a repeated root), or no real solutions.

Methods for Solving Quadratic Equations

Several methods exist for solving quadratic equations. We'll explore the most common and effective techniques, applying them to our example, 4x² + 5x - 12 = 0.

1. Factoring

Factoring involves rewriting the quadratic equation as a product of two linear expressions. This method is efficient when the equation can be easily factored. For 4x² + 5x - 12 = 0, we look for two numbers that add up to 5 (the coefficient of x) and multiply to -48 (the product of a and c, which is 4 * -12). These numbers are 8 and -3.

We can rewrite the equation as:

4x² + 8x - 3x - 12 = 0

Now, factor by grouping:

4x(x + 2) - 3(x + 2) = 0

(4x - 3)(x + 2) = 0

This gives us two possible solutions:

• 4x - 3 = 0 => x = 3/4
• x + 2 = 0 => x = -2

Therefore, the solutions to 4x² + 5x - 12 = 0 are x = 3/4 and x = -2.

2. Quadratic Formula

The quadratic formula provides a universal solution for any quadratic equation, regardless of its factorability. The formula is derived from completing the square and is given by:

x = [-b ± √(b² - 4ac)] / 2a

For our equation, 4x² + 5x - 12 = 0, we have a = 4, b = 5, and c = -12. Substituting these values into the quadratic formula:

x = [-5 ± √(5² - 4 * 4 * -12)] / (2 * 4)

x = [-5 ± √(25 + 192)] / 8

x = [-5 ± √217] / 8

This gives us two solutions:

• x = (-5 + √217) / 8 ≈ 1.136
• x = (-5 - √217) / 8 ≈ -2.636

Note that these solutions are approximate due to the irrational nature of √217. These approximate solutions are close to the exact solutions we obtained through factoring (3/4 and -2), highlighting the consistency between methods.

3. Completing the Square

Completing the square is a method that involves manipulating the equation to create a perfect square trinomial. This method is particularly useful when the quadratic equation is not easily factorable. While less straightforward than the quadratic formula, it provides a deeper understanding of the underlying algebraic manipulations.

Let's apply completing the square to 4x² + 5x - 12 = 0:

1. Divide by 'a': x² + (5/4)x - 3 = 0

2. Move the constant to the right side: x² + (5/4)x = 3

3. Complete the square: To complete the square, take half of the coefficient of x ((5/4)/2 = 5/8), square it ((5/8)² = 25/64), and add it to both sides:

x² + (5/4)x + 25/64 = 3 + 25/64

4. Factor the perfect square trinomial: (x + 5/8)² = 217/64

5. Solve for x: x + 5/8 = ±√(217/64)

x = -5/8 ± √217/8

This yields the same solutions as the quadratic formula.

Discriminant and Nature of Roots

The discriminant, denoted by Δ (delta), is the expression inside the square root in the quadratic formula: b² - 4ac. The discriminant determines the nature of the roots:

• Δ > 0: Two distinct real roots (as in our example using the quadratic formula).
• Δ = 0: One repeated real root (a single solution).
• Δ < 0: No real roots (the roots are complex numbers involving 'i', the imaginary unit).

For 4x² + 5x - 12 = 0, the discriminant is:

Δ = 5² - 4 * 4 * -12 = 217 > 0

This confirms that the equation has two distinct real roots.

Applications of Quadratic Equations

Quadratic equations are not merely abstract mathematical concepts; they have widespread applications in diverse fields:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity.
• Engineering: Designing bridges, analyzing structural stability, and optimizing systems.
• Finance: Modeling compound interest, determining investment returns, and analyzing market trends.
• Computer Science: Creating algorithms, solving optimization problems, and developing graphical interfaces.
• Economics: Modeling supply and demand curves, understanding market equilibrium, and optimizing resource allocation.

Graphical Representation

The graph of a quadratic equation is a parabola. The x-intercepts of the parabola represent the roots (solutions) of the equation. For 4x² + 5x - 12 = 0, the parabola intersects the x-axis at x = 3/4 and x = -2, visually confirming our calculated solutions. The parabola opens upwards because the coefficient of x² (a = 4) is positive.

Conclusion: Mastering Quadratic Equations

Solving quadratic equations is a fundamental skill in mathematics with far-reaching implications. This in-depth exploration of 4x² + 5x - 12 = 0 demonstrated various solution methods – factoring, the quadratic formula, and completing the square – highlighting their respective strengths and applications. Understanding the discriminant allows us to predict the nature of the roots without explicitly solving the equation. The versatility and applicability of quadratic equations make mastering them a crucial step towards advanced mathematical understanding and problem-solving capabilities across multiple disciplines. Continued practice and exploration of different problem types will solidify your understanding and enhance your ability to confidently tackle more complex mathematical challenges.