Solving 49 + 16x(4x + 7) = 0 With The Quadratic Formula A Step-by-Step Guide

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In this comprehensive guide, we will delve into the process of solving the quadratic equation 49 + 16x(4x + 7) = 0 using the renowned Quadratic Formula. This formula is a cornerstone in algebra, providing a systematic approach to finding the roots (or solutions) of any quadratic equation. Our exploration will not only demonstrate the application of the formula but also provide a step-by-step breakdown to ensure clarity and understanding. We will emphasize the importance of precision in calculations and the significance of interpreting the results in the context of mathematical problem-solving. Whether you're a student grappling with algebra or simply seeking to refresh your mathematical skills, this guide will equip you with the knowledge and confidence to tackle quadratic equations effectively. Our journey will begin with understanding the standard form of a quadratic equation and identifying the coefficients necessary for the Quadratic Formula. Then, we will meticulously apply the formula, paying close attention to each step to avoid common errors. Finally, we will interpret the solutions, discussing the nature of the roots (real or complex) and their implications. This methodical approach will not only help in solving the specific equation at hand but also lay a solid foundation for tackling more complex mathematical problems in the future.

Understanding the Quadratic Formula

The Quadratic Formula is a powerful tool for solving quadratic equations, which are equations of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. This formula is particularly useful when factoring the quadratic equation is difficult or impossible. The Quadratic Formula itself is expressed as:

x = (-b ± √(b² - 4ac)) / (2a)

Where:

  • x represents the solutions or roots of the quadratic equation.
  • a is the coefficient of the x² term.
  • b is the coefficient of the x term.
  • c is the constant term.
  • The symbol ± indicates that there are two possible solutions, one with addition and one with subtraction.
  • The expression b² - 4ac under the square root is known as the discriminant. The discriminant plays a crucial role in determining the nature of the roots:
    • If b² - 4ac > 0, the equation has two distinct real roots.
    • If b² - 4ac = 0, the equation has one real root (a repeated root).
    • If b² - 4ac < 0, the equation has two complex roots.

Understanding the Quadratic Formula is essential for solving a wide range of mathematical problems, from simple algebraic equations to more complex applications in physics and engineering. The formula provides a direct method for finding the roots of any quadratic equation, regardless of the complexity of the coefficients. This is why it is a fundamental concept in algebra and a vital tool for anyone working with mathematical equations. Before we can apply the Quadratic Formula, we need to ensure that our equation is in the standard form ax² + bx + c = 0. This involves rearranging the terms if necessary and identifying the coefficients a, b, and c. Once we have these values, we can substitute them into the formula and perform the calculations to find the solutions. The process might seem daunting at first, but with practice, it becomes a straightforward and reliable method for solving quadratic equations.

Step-by-Step Solution for 49 + 16x(4x + 7) = 0

To solve the quadratic equation 49 + 16x(4x + 7) = 0 using the Quadratic Formula, we'll follow a structured, step-by-step approach. This method ensures accuracy and clarity in our solution. Let's break down the process:

Step 1: Convert the Equation to Standard Form

The first crucial step is to rewrite the given equation in the standard quadratic form, which is ax² + bx + c = 0. The original equation is 49 + 16x(4x + 7) = 0. We need to expand and simplify this equation:

  1. Distribute the 16x across the terms inside the parentheses: 16x * 4x = 64x² and 16x * 7 = 112x.
  2. The equation now becomes: 49 + 64x² + 112x = 0.
  3. Rearrange the terms to match the standard form: 64x² + 112x + 49 = 0.

Now, the equation is in the standard form, making it easier to identify the coefficients for the Quadratic Formula.

Step 2: Identify the Coefficients a, b, and c

In the standard form ax² + bx + c = 0, we can now identify the values of a, b, and c from our equation 64x² + 112x + 49 = 0:

  • a is the coefficient of the x² term, which is 64.
  • b is the coefficient of the x term, which is 112.
  • c is the constant term, which is 49.

These values are essential for plugging into the Quadratic Formula.

Step 3: Apply the Quadratic Formula

Now that we have identified a = 64, b = 112, and c = 49, we can substitute these values into the Quadratic Formula:

x = (-b ± √(b² - 4ac)) / (2a)

Substitute the values:

x = (-112 ± √(112² - 4 * 64 * 49)) / (2 * 64)

Step 4: Simplify the Expression

Let's simplify the expression step by step:

  1. Calculate b²: 112² = 12544.

  2. Calculate 4ac: 4 * 64 * 49 = 12544.

  3. Substitute these values back into the formula:

    x = (-112 ± √(12544 - 12544)) / (128)

  4. Simplify the expression under the square root:

    x = (-112 ± √0) / (128)

  5. Since the square root of 0 is 0, the equation becomes:

    x = (-112 ± 0) / (128)

Step 5: Determine the Solutions

Since the discriminant (b² - 4ac) is 0, there is only one real solution (a repeated root):

x = -112 / 128

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 16:

x = -7 / 8

Therefore, the solution to the quadratic equation 49 + 16x(4x + 7) = 0 is x = -7/8.

This step-by-step solution demonstrates the application of the Quadratic Formula in a clear and concise manner. By following these steps, you can confidently solve similar quadratic equations.

Interpreting the Solution and the Discriminant

After successfully applying the Quadratic Formula to solve the equation 49 + 16x(4x + 7) = 0, we arrived at the solution x = -7/8. However, the journey doesn't end with just finding the numerical answer. A crucial aspect of mathematical problem-solving is understanding the meaning of the solution and the information provided by the discriminant.

In our case, the discriminant (b² - 4ac) played a significant role in determining the nature of the roots. Let's revisit the discriminant and its implications:

  • Discriminant (b² - 4ac) = 112² - 4 * 64 * 49 = 12544 - 12544 = 0

A discriminant of 0 indicates that the quadratic equation has exactly one real root, which is a repeated root. This means that the parabola represented by the quadratic equation touches the x-axis at only one point, which corresponds to our solution x = -7/8.

Interpreting the Solution:

The solution x = -7/8 represents the x-coordinate of the point where the parabola intersects the x-axis. Since there is only one solution, the vertex of the parabola lies on the x-axis. This understanding is vital for visualizing the graph of the quadratic equation and gaining a deeper insight into its behavior.

The Significance of the Discriminant:

The discriminant is a powerful tool for quickly determining the nature of the roots without fully solving the equation. Here's a summary of its implications:

  • If b² - 4ac > 0: The equation has two distinct real roots. The parabola intersects the x-axis at two different points.
  • If b² - 4ac = 0: The equation has one real root (a repeated root). The vertex of the parabola lies on the x-axis.
  • If b² - 4ac < 0: The equation has two complex roots. The parabola does not intersect the x-axis.

In conclusion, interpreting the solution and understanding the discriminant are essential steps in solving quadratic equations. They provide valuable insights into the nature of the roots and the graphical representation of the equation. This comprehensive understanding enhances problem-solving skills and allows for a more profound appreciation of mathematical concepts.

Common Mistakes to Avoid When Using the Quadratic Formula

While the Quadratic Formula is a reliable method for solving quadratic equations, it's crucial to be aware of common mistakes that can occur during its application. Avoiding these pitfalls ensures accuracy and efficiency in problem-solving. Here are some key errors to watch out for:

  1. Incorrectly Identifying Coefficients:

    • Mistake: Misidentifying the values of a, b, and c from the standard form ax² + bx + c = 0. This is a frequent error, especially when the equation is not initially in standard form.
    • How to Avoid: Always rewrite the equation in standard form before identifying the coefficients. Pay close attention to the signs (positive or negative) of the coefficients.

  2. Sign Errors:

    • Mistake: Making errors with the signs, particularly when substituting negative values for b in the formula x = (-b ± √(b² - 4ac)) / (2a).
    • How to Avoid: Use parentheses when substituting values into the formula, especially for negative numbers. For instance, if b is -5, write -(-5) in the formula to avoid sign errors.

  3. Calculation Errors Under the Square Root:

    • Mistake: Incorrectly calculating the discriminant (b² - 4ac). This can lead to determining the wrong number of roots or the wrong type of roots (real or complex).
    • How to Avoid: Perform the calculations under the square root carefully, following the order of operations (PEMDAS/BODMAS). Double-check your arithmetic to ensure accuracy.

  4. Forgetting the ± Sign:

    • Mistake: Neglecting the ± sign in the Quadratic Formula, which indicates two possible solutions.
    • How to Avoid: Remember that the ± sign represents both addition and subtraction, leading to two potential roots (unless the discriminant is 0). Always consider both possibilities.

  5. Dividing Only Part of the Numerator:

    • Mistake: Dividing only one term in the numerator by 2a instead of dividing the entire numerator.
    • How to Avoid: Ensure that the entire numerator (-b ± √(b² - 4ac)) is divided by 2a. It can be helpful to use parentheses to emphasize this division.

  6. Simplifying Too Early:

    • Mistake: Attempting to simplify the expression before completing the calculations under the square root.
    • How to Avoid: Calculate the discriminant (b² - 4ac) first, then simplify the square root before attempting any further simplification.

By being mindful of these common mistakes and implementing the suggested strategies, you can significantly improve your accuracy and confidence when using the Quadratic Formula. Practice and attention to detail are key to mastering this essential algebraic tool.

Real-World Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications across various fields. Understanding these applications can provide a deeper appreciation for the practical significance of quadratic equations and the Quadratic Formula. Let's explore some key areas where quadratic equations play a crucial role:

  1. Physics:

    • Projectile Motion: Quadratic equations are fundamental in describing the trajectory of projectiles, such as balls thrown in the air or rockets launched into space. The equation of motion often involves quadratic terms that account for the effect of gravity and air resistance.
    • Kinematics: They are used to model the relationship between displacement, velocity, and acceleration in scenarios where acceleration is constant. For example, calculating the time it takes for a car to stop given its initial speed and deceleration rate involves solving a quadratic equation.

  2. Engineering:

    • Structural Engineering: Quadratic equations are used in designing structures like bridges and buildings. Engineers need to calculate the forces and stresses acting on different parts of the structure, which often involves solving quadratic equations to ensure stability and safety.
    • Electrical Engineering: In circuit analysis, quadratic equations can help determine the current and voltage in circuits containing resistors, capacitors, and inductors. These equations are essential for designing and optimizing electrical systems.

  3. Computer Graphics:

    • Curve Modeling: Quadratic equations are used to create curves and surfaces in computer graphics. Bezier curves, which are widely used in graphic design and animation, are based on quadratic and cubic polynomials. These curves allow for smooth and realistic shapes to be rendered on screens.
    • Collision Detection: In video games and simulations, quadratic equations can be used to detect collisions between objects. By modeling the objects' paths using quadratic equations, developers can determine when and where collisions occur.

  4. Economics and Finance:

    • Optimization Problems: Quadratic functions are used to model cost, revenue, and profit functions. Businesses can use quadratic equations to find the optimal production levels or pricing strategies that maximize profit or minimize cost.
    • Financial Modeling: Quadratic equations can be used to model investment growth and depreciation. For example, calculating the time it takes for an investment to reach a certain value, given a compound interest rate, can involve solving a quadratic equation.

  5. Everyday Life:

    • Gardening and Landscaping: Quadratic equations can be used to calculate the dimensions of a rectangular garden with a specific area, or to determine the amount of fencing needed for a yard.
    • Home Improvement: When planning home renovations, quadratic equations can help calculate the amount of materials needed for projects such as tiling a floor or building a deck.

These examples illustrate the wide-ranging applicability of quadratic equations in various fields. From designing safe structures to optimizing financial strategies, quadratic equations provide a powerful tool for solving real-world problems. The Quadratic Formula is not just a mathematical abstraction but a practical tool that helps us understand and shape the world around us.

In summary, mastering the art of solving quadratic equations using the Quadratic Formula is an invaluable skill. Through a meticulous step-by-step approach, careful attention to detail, and an understanding of the discriminant, one can confidently tackle a wide array of mathematical challenges. Furthermore, recognizing the real-world applications of these equations enriches our appreciation for their significance and utility. By avoiding common pitfalls and embracing the power of the Quadratic Formula, we unlock a fundamental tool for problem-solving in mathematics and beyond.