Solving the Equation: 4x ^ 2 - 5x - 12 = 0

Quadratic equations, which have the general form $a x^{2} + b x + c = 0$ , where $a$ , $b$ , and $c$ are constants and a is not equals to 0 are fundamental to algebra and appear frequently in various mathematical and real-world applications. Today, we delve into solving and understanding the characteristics of a specific quadratic equation: 4x ^ 2 – 5x – 12 = 0.

Solving the Equation 4x ^ 2 – 5x – 12 = 0

To find the roots of the equation 4x ^ 2 – 5x – 12 = 0, we can use the quadratic formula, which is given by:

x = \frac{-b \pm \sqrt{b^{2} - 4 ac}}{2 a}

In our equation, $a = 4$ , $b = - 5$ , and $c = - 12$ . Substituting these values into the formula gives us:

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

Let’s compute the exact values of $x$ .

Calculation

Plugging the values into the formula:

x = \frac{5 \pm \sqrt{25 + 192}}{8} = \frac{5 \pm \sqrt{217}}{8​​}

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

x_{1} = \frac{5 + \sqrt{217}}{8},x_{2} = \frac{5 - \sqrt{217}}{8​}

These roots represent the x-coordinates where the parabola, represented by the quadratic equation, intersects the x-axis.

Graphical Representation

Graphically, a quadratic equation represents a parabola. For our equation, the parabola opens upwards (since $a = 4$ is positive), and its vertex can be found using the formula:

x_{v} = \frac{-b}{2 a}, y_{v} = c - \frac{b^{2}}{4 a​}

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Applications and Implications

Quadratic equations like 4x ^ 2 – 5x – 12 = 0 are not just academic exercises; they have practical applications in physics, engineering, economics, and various other fields. For example, they can represent the trajectory of projectiles, the optimization of profit in business models, or the calculation of areas and dimensions in architectural designs.

Understanding how to solve such equations and interpret their solutions allows us to predict, optimize, and model numerous real-world phenomena. The roots of the equation provide key insights into the behavior of the system being modeled, such as the maximum height reached by a projectile or the break-even points in a business scenario.

Conclusion

The quadratic equation 4x ^ 2 – 5x – 12 = 0 serves as a clear example of the power and applicability of algebra in solving real-world problems. By analyzing and solving this equation, we gain not only the ability to find its roots but also deeper insights into the characteristics and applications of quadratic functions. Whether in academic settings or real-world scenarios, the skills developed through such exercises are invaluable.

Solving the Equation: 4x ^ 2 – 5x – 12 = 0