Solving Quadratic Equations 6x^2 - 2x - 4 = 0 With The Quadratic Formula

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In the realm of mathematics, quadratic equations hold a significant position, appearing in various applications from physics to engineering. One of the most powerful tools for solving these equations is the quadratic formula. This article provides a comprehensive guide on how to use the quadratic formula to solve the equation $6x^2 - 2x - 4 = 0$, ensuring clarity and understanding for learners of all levels.

Understanding Quadratic Equations

Before diving into the solution, let's establish a clear understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (typically 'x') is 2. The standard form of a quadratic equation is expressed as:

ax2+bx+c=0ax^2 + bx + c = 0

where 'a', 'b', and 'c' are coefficients, with 'a' not equal to zero. These coefficients play a crucial role in determining the solutions or roots of the equation. The solutions represent the values of 'x' that satisfy the equation, making the expression equal to zero.

Key characteristics of quadratic equations include:

  • Two Solutions: Quadratic equations generally have two solutions, which may be real or complex numbers. These solutions can be distinct, repeated, or complex conjugates.
  • Parabolic Graph: When plotted on a graph, a quadratic equation forms a parabola, a U-shaped curve. The solutions of the equation correspond to the x-intercepts of the parabola, where the curve intersects the x-axis.
  • Various Solution Methods: Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula. Each method has its strengths and weaknesses, and the choice of method often depends on the specific equation and personal preference.

Understanding these fundamental aspects of quadratic equations is crucial for effectively applying the quadratic formula and interpreting the solutions obtained.

The Quadratic Formula: A Universal Solution

The quadratic formula is a powerful tool that provides a universal solution for any quadratic equation. It states that for an equation in the standard form $ax^2 + bx + c = 0$, the solutions for 'x' are given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula may seem intimidating at first, but it is a direct result of the method of completing the square applied to the general quadratic equation. Let's break down the components of the formula:

  • -b: The negative of the coefficient 'b'.
  • ±: This symbol indicates that there are two possible solutions, one with addition and one with subtraction.
  • √(b² - 4ac): The square root of the discriminant (b² - 4ac). The discriminant determines the nature of the solutions (real or complex).
  • b² - 4ac: The discriminant, a crucial part of the formula. It reveals the nature and number of solutions:
    • If b² - 4ac > 0: Two distinct real solutions.
    • If b² - 4ac = 0: One real solution (a repeated root).
    • If b² - 4ac < 0: Two complex solutions.
  • 2a: Twice the coefficient 'a'.

The quadratic formula is a versatile tool because it can be applied to any quadratic equation, regardless of whether it can be easily factored or solved by other methods. It is a cornerstone of algebra and a must-know for anyone working with quadratic equations.

Solving $6x^2 - 2x - 4 = 0$ Using the Quadratic Formula

Now, let's apply the quadratic formula to solve the given equation: $6x^2 - 2x - 4 = 0$. This step-by-step approach will make the application of the formula clear and understandable.

1. Identify the Coefficients

First, we need to identify the coefficients 'a', 'b', and 'c' from the equation $6x^2 - 2x - 4 = 0$. By comparing this equation to the standard form $ax^2 + bx + c = 0$, we can easily identify the coefficients:

  • a = 6
  • b = -2
  • c = -4

2. Substitute the Coefficients into the Formula

Next, we substitute these values into the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Substituting a = 6, b = -2, and c = -4, we get:

x=−(−2)±(−2)2−4(6)(−4)2(6)x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(6)(-4)}}{2(6)}

This step involves careful substitution to avoid errors and ensure the correct application of the formula.

3. Simplify the Expression

Now, we simplify the expression step by step:

x=2±4+9612x = \frac{2 \pm \sqrt{4 + 96}}{12}

x=2±10012x = \frac{2 \pm \sqrt{100}}{12}

x=2±1012x = \frac{2 \pm 10}{12}

This simplification process involves basic arithmetic operations and the evaluation of the square root. It's crucial to follow the order of operations (PEMDAS/BODMAS) to ensure accurate simplification.

4. Calculate the Two Solutions

Finally, we calculate the two solutions by considering both the plus and minus signs:

  • Solution 1 (using +):

    x1=2+1012=1212=1x_1 = \frac{2 + 10}{12} = \frac{12}{12} = 1

  • Solution 2 (using -):

    x2=2−1012=−812=−23x_2 = \frac{2 - 10}{12} = \frac{-8}{12} = -\frac{2}{3}

Therefore, the solutions to the quadratic equation $6x^2 - 2x - 4 = 0$ are x = 1 and x = -2/3. These values of 'x' satisfy the original equation, making the expression equal to zero.

Verifying the Solutions

To ensure the accuracy of our solutions, it's a good practice to verify them by substituting them back into the original equation. This step helps to identify any potential errors made during the solution process.

Verifying x = 1

Substituting x = 1 into the equation $6x^2 - 2x - 4 = 0$, we get:

6(1)2−2(1)−4=6−2−4=06(1)^2 - 2(1) - 4 = 6 - 2 - 4 = 0

Since the result is 0, x = 1 is indeed a valid solution.

Verifying x = -2/3

Substituting x = -2/3 into the equation $6x^2 - 2x - 4 = 0$, we get:

6(−23)2−2(−23)−4=6(49)+43−46(-\frac{2}{3})^2 - 2(-\frac{2}{3}) - 4 = 6(\frac{4}{9}) + \frac{4}{3} - 4

=249+43−4=83+43−123=0= \frac{24}{9} + \frac{4}{3} - 4 = \frac{8}{3} + \frac{4}{3} - \frac{12}{3} = 0

Since the result is also 0, x = -2/3 is a valid solution as well.

By verifying both solutions, we can be confident that our calculations are correct and that we have found the accurate roots of the quadratic equation.

Real-World Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have numerous applications in the real world. Understanding these applications can help to appreciate the significance of quadratic equations and their solutions.

Physics:

  • Projectile Motion: The trajectory of a projectile, such as a ball thrown in the air, can be modeled using a quadratic equation. The equation can predict the height and range of the projectile, considering factors like initial velocity and launch angle.
  • Simple Harmonic Motion: The motion of an object oscillating back and forth, like a pendulum or a spring, can be described by a quadratic equation. The equation can determine the position and velocity of the object at any given time.

Engineering:

  • Structural Design: Quadratic equations are used to calculate the stress and strain on structural components, such as beams and bridges. This ensures that structures are strong enough to withstand loads and remain stable.
  • Circuit Analysis: Quadratic equations are used to analyze electrical circuits, determining the current and voltage in various components. This is essential for designing efficient and reliable electrical systems.

Economics:

  • Cost-Benefit Analysis: Quadratic equations can be used to model cost and revenue functions, helping businesses to determine optimal production levels and pricing strategies.
  • Financial Modeling: Quadratic equations can be used to analyze investment returns and calculate interest rates, aiding in financial planning and decision-making.

These are just a few examples of the many real-world applications of quadratic equations. Their ability to model various phenomena makes them an essential tool in many fields.

Conclusion

The quadratic formula is an indispensable tool for solving quadratic equations. By understanding the formula and its components, and by following a step-by-step approach, anyone can confidently solve quadratic equations. This article has provided a detailed guide on how to use the quadratic formula to solve the equation $6x^2 - 2x - 4 = 0$, along with explanations of quadratic equations, the formula itself, and real-world applications. With practice and a solid understanding of the concepts, you can master the quadratic formula and its applications.

Remember, mathematics is not just about formulas and equations; it's about understanding the underlying concepts and applying them to solve problems. So, keep practicing, keep exploring, and keep learning!