Solve: 4x ^ 2 – 5x – 12 = 0

Quadratic equations form the cornerstone of algebra, opening doors to various fields of mathematics and science. Among these, the equation 4x ^ 2 – 5x – 12 = 0 stands as a quintessential example, showcasing the beauty and complexity of algebraic solutions. This article embarks on a detailed exploration of solving this equation, providing readers with a comprehensive understanding and the tools necessary to tackle similar problems.

Understanding Quadratic Equations

Before diving into the solution, it’s imperative to grasp what quadratic equations are. In essence, a quadratic equation is any equation that can be rearranged into the standard form ax2+bx+c=0, where $$, $b$, and $c$ are constants, and $$ is not equal to zero. The quadratic equation 4x ^ 2 – 5x – 12 = 0 is a prime example, with $a=4$, $$, and $c=-12$.

Methodical Approach to Solving 4x ^ 2 – 5x – 12 = 0

Step 1: Identify the Coefficients

The first step in solving any quadratic equation is identifying the coefficients and constant term. For $4{�}^2-5�-12=0$, we have:

• $�=4$
• $�=-5$
• $�=-12$

Step 2: The Quadratic Formula

The quadratic formula, $x=\frac{-b\pm \sqrt{b^2-4\mathrm{ac}}}{2a}$, is a powerful tool for solving quadratic equations. Applying this formula to our equation, we substitute the identified values of $a$, $$, and $$.

Step 3: Calculation

Substituting $\mathrm{}$, $$, and $$ into the quadratic formula gives us:

$x=\frac{-(-5)\pm \sqrt{(-5{)}^2-4(4)(-12)}}{2(4)}$

$x=\frac{5\pm \sqrt{25+192}}{8}$

$x=\frac{5\pm \sqrt{217}}{8}$

Thus, the solutions to 4x ^ 2 – 5x – 12 = 0 are:

$x=\frac{5+\sqrt{217}}{8}$

$x=\frac{5-\sqrt{217}}{8}$

Understanding the Solutions

The solutions to 4x ^ 2 – 5x – 12 = 0 are not just numbers; they are the roots of the equation, where the graph of 4x ^ 2 – 5x – 12 = 0 intersects the x-axis. These roots are crucial for various applications, including physics, engineering, and finance.

Applying the Knowledge

Understanding how to solve quadratic equations like 4x ^ 2 – 5x – 12 = 0 equips learners with essential mathematical skills. Whether it’s for academic purposes, professional applications, or personal curiosity, mastering this process is invaluable.

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Conclusion

The equation 4x ^ 2 – 5x – 12 = 0 is more than just a mathematical problem; it’s a gateway to understanding the fundamentals of algebra. Through detailed steps and explanations, we’ve unraveled the complexities of solving quadratic equations, providing a solid foundation for tackling even more challenging problems.

Quadratic equations are a testament to the beauty and applicability of mathematics. By demystifying their solutions, we not only enhance our problem-solving skills but also appreciate the intricate connections between mathematical concepts and the real world.

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